UNIVERSITY OF CALIFORNIA, SAN DIEGO
A New Ultra-Cold Positron Beam and Applications
To Low-Energy Positron Scattering and
Electron-Positron Plasmas
A dissertation submitted in partial satisfaction of the requirements for
the degree of Doctor of Philosophy in Physics
by
Steven Jay Gilbert
Committee in charge:
Cliord M. Surko, Chair
Robert Continetti
Andrew Kummel
Thomas O'Neil
Arthur Wolfe
2000
Copyright
Steven Jay Gilbert, 2000
All rights reserved.
The dissertation of Steven Jay Gilbert is approved and it is acceptable in
quality and form for publication on microlm.
Chairman
University of California, San Diego
2000
iii
This thesis is dedicated to my life-mate, Anne.
iv
Contents
Signature Page . . . . . . . . . . . . .
Table of Contents . . . . . . . . . . . .
List of Figures . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . .
Vita, Publications and Fields of Study
Abstract . . . . . . . . . . . . . . . . .
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1
Introduction
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General Description of the Experiment
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Bright, Cold Charged Particle Beams
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Positron Scattering from Atoms and Molecules
1.1
1.2
1.3
1.4
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4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Experimental Setup . . . . . . . . . . . . . . . . . . .
4.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Elastic DCS Analysis . . . . . . . . . . . . . .
4.3.2 Measurement of Total Inelastic Cross Sections
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2.1
2.2
2.3
2.4
2.5
3.1
3.2
3.3
3.4
3.5
Positron Sources . . . . . . . . . . . . . . . . . .
Moderators to Produce Low-Energy Positrons . .
Positron Physics at Lower Energies . . . . . . . .
Overview of the Thesis . . . . . . . . . . . . . . .
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Charged Particle Motion in a Magnetic Field
Source and Moderator . . . . . . . . . . . . .
Beam Extraction and Transport . . . . . . .
Positron Accumulation . . . . . . . . . . . . .
Retarding Potential Analyzer (RPA) . . . . .
Introduction . . . . .
Experimental Setup
Positron Beams . . .
Electron Beams . . .
Chapter Summary .
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4.3.3 Total Cross Sections . . . . . . . . . . . . . . . . . . . . .
4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Dierential Cross Sections . . . . . . . . . . . . . . . . .
4.4.2 Total Inelastic Cross Sections . . . . . . . . . . . . . . . .
4.4.3 Recent Results . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Measurements Using a Magnetic Beam { Further Considerations
4.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
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The Electron-Positron Beam-Plasma Instability
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Conclusion
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5.1 Introduction . . . . . . . . . . . . . . . .
5.2 Description of the Experiment . . . . . .
5.2.1 Positron Plasma Parameters . .
5.2.2 Cold Electron Beam Parameters
5.2.3 Beam-Plasma Experiment . . . .
5.3 Results . . . . . . . . . . . . . . . . . . .
5.4 Chapter Summary . . . . . . . . . . . .
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6.1 Summary . . . . . . . . . . . . . . . . . . . . . . .
6.2 Future Work . . . . . . . . . . . . . . . . . . . . .
6.2.1 Cold Beams . . . . . . . . . . . . . . . . . .
6.2.2 Positron Atomic and Molecular Scattering .
6.2.3 Electron-Positron Plasmas . . . . . . . . . .
6.3 Concluding Remarks . . . . . . . . . . . . . . . . .
References
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List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
3.1
3.2
3.3
3.4
3.5
3.6
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
5.1
Charged particle motion in a magnetic eld . . . . . . . . . . . .
Schematic diagram of the positron source and moderator . . . . .
Schematic diagram of the positron source vacuum chamber . . .
Schematic diagram of the three-stage positron accumulator . . .
Buer gas pressure in the third stage during pump out . . . . . .
Calculated pressure prole along the three-stage accumulator . .
Schematic diagram showing the three-stage accumulator and surrounding vacuum chamber . . . . . . . . . . . . . . . . . . . . . .
Energy distribution of a moderated positron beam . . . . . . . .
Schematic diagram of the beam formation experiment . . . . . .
Pulse train of 60 positron pulses . . . . . . . . . . . . . . . . . .
Energy distribution of a positron pulse . . . . . . . . . . . . . . .
Potential distribution in cylindrical trap . . . . . . . . . . . . . .
Energy distribution of a 0.1 A quasi steady-state electron beam
Current waveform of an electron beam . . . . . . . . . . . . . . .
Schematic diagram of scattering apparatus and pressure prole .
Scattering in a magnetic eld . . . . . . . . . . . . . . . . . . . .
Simulated eects of scattering on the beam energy . . . . . . . .
RPA data for positron-argon elastic scattering . . . . . . . . . . .
Dierential elastic cross sections for positron-krypton scattering .
Dierential elastic cross sections for positron-argon scattering . .
RPA data for a positron-CF4 inelastic scattering event . . . . . .
Positron-CF4 inelastic cross sections . . . . . . . . . . . . . . . .
Positron-CO inelastic cross sections . . . . . . . . . . . . . . . . .
Positron-CH4 inelastic cross sections . . . . . . . . . . . . . . . .
Positron-CO2 inelastic cross sections . . . . . . . . . . . . . . . .
Total cross sections for positron CF4 , CH3F, and CH4 . . . . . .
Experimental limitations . . . . . . . . . . . . . . . . . . . . . . .
Eects of time resolving the DCS measurement . . . . . . . . . .
Schematic diagram of the beam-plasma experiment . . . . . . . .
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5.2
5.3
5.4
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Beam and plasma radial density proles . . . . . . . . . . . . .
Density distribution of the positron plasma . . . . . . . . . . .
Potential prole along the quadrupole traps axial symmetry . .
RMS data showing the excitation of the transit-time instability
Growth rates for the beam-plasma instability . . . . . . . . . .
viii
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Acknowledgments
I would like to thank all of the people who helped me through the ups and
downs of my graduate career.
It goes without saying that the single most inuential person in shaping my
progress through the graduate program was my advisor, Profesor Cli Surko.
The process of choosing a graduate advisor is part research, part intuition, and
part luck. I sought out someone who could guide me through all of the diÆculties
of a graduate research program. Cli turned out to be an excellent choice in
every respect. He has subtly taught me over the years how to get through, over,
or around the many brick walls which arise during the experimental process.
Along with Cli, I am indebted to all of the people I had the pleasure of
working with in his lab. Thanks to Greg, Christof, Koji, James, Gene, Rod,
Chris Lund, Chris Kurz, and Judy. Special thanks to Chris Kurz, a wonderful mentor who introduced me to the art of experimental research, and helped
greatly on the work presented in this thesis relating to the cold-beam formation.
Researcher Rod Greaves oered his keen insight into both experimental design
and the physical processes at work. Thanks to him I have gained condence in
my ability to design, build, and implement an experiment from the ground up.
None of this work could have been done without Gene Jerzewski's expert
technical assistance. He is a true Zen master of the lab. Despite the many
attempts of the lab to break down, he was always there with a smile to tell me
not to worry, that it could be xed, and he was always right. For the last year
James Sullivan has educated me in the ways of an atomic-physicist. He has also
helped to take much of the recent data presented in Chapter 4, while I have been
hunched over the computer writing this dissertation.
On a personal note, I would like to thank my family and friends for keeping
me mostly sane through it all. Thanks to Mom, Eric, Dad, Pauline, Dennis,
Nancy, Debbie, Mike, Stefanie, Jordan, Nan, and Ken, who complete one of the
greatest families ever. Most of all I would like to thank Anne, for sticking so
close to me through the thick and thin of it all. She has been a great source of
inspiration and the Ph.D. degree is as much hers as it is mine. I would also like
to thank my mother for teaching me that the day to day of work was something
to be enjoyed not wished away until the next weekend or vacation. She also
never discouraged me from asking why, no matter how many frustrating times I
posed the question, and this is what led me into the sciences. I owe my sense of
perspective to my brothers and sisters, although we have each pursued dierent
things to become happy, your successes have inspired me.
An integral aspect of my life has been my love of the outdoors which originated during the many walks through the woods taken with my step-farther
Eric. Whether I am hiking, skiing, climbing, or biking, he will always be part of
me when I am in the woods. Even though Eric started my love of the outdoors,
he would never have pushed me towards rock-climbing. For that I owe my good
ix
friend Todd, thanks for being a great teacher. Since I have been in San Diego,
the Thursday-Sunday climbing group has been responsible for an uncountable
number of great times outdoors. Thanks Doug, Garnet, Cindy, Frank, Mike,
and all of the other wonderful people in the group.
Lastly, I would like to thank the friends I have made throughout my college and graduate career for all the good times that they have given me. At
the University of Rochester, I met Dylan and learned that being yourself, and
remaining forever a kid was something worth striving for. At Rutgers, Amy
started me on the way to making a number of life-choices, including becoming
a vegetarian. Since I have been at the University of California, San Diego, my
friends Thor, Neda, and Jason, were rst my study partners, and later my commiseration partners, as we all worked our way through the graduate process.
Finally, thanks to Ray for sharing his enthusiasm for life, always with a stupid
joke on hand.
x
Vita
May 12, 1970
1993
1994
1995
2000
Born, New Jersey, USA.
B.A., Physics, Rutgers University.
Research Assistant, University of California, San Diego.
M.S., Physics, University of California, San Diego.
Ph.D., Physics, University of California, San Diego.
Publications
ARTICLES
1. S. J. Gilbert, D. H. E. Dubin, R. G. Greaves, and C. M. Surko, \An
Electron Positron Beam-Plasma Instability," manuscript in preparation.
2. S. J. Gilbert, J. Sullivan, R. G. Greaves, and C. M. Surko, \Low Energy
Positron Scattering from Atoms and Molecules using Positron Accumulation Techniques," Nuclear Instruments and Methods in Physics Research
B, in press.
3. S. J. Gilbert, R. G. Greaves, and C. M. Surko, \Positron Scattering from
Atoms and Molecules at Low Energies," Physical Review Letters , 50325035 (1999).
4. C. Kurz, S. J. Gilbert, R. G. Greaves, and C. M. Surko, \New Source
of Ultra-Cold Positron and Electron Beams," Nuclear Instruments and
Methods in Physics Research B
, 188-194 (1998).
5. S. J. Gilbert, C. Kurz, R. G. Greaves, and C. M. Surko, \Creation of a
monoenergetic pulsed positron beam," Applied Physics Letters , 19441946 (1997).
6. K. Iwata, R. G. Greaves, C. Kurz, S. J. Gilbert and C. M. Surko, \Studies
of positron-matter interactions using stored positrons in an electrostatic
trap," Materials Science Forum , 223-227 (1997).
7. C. M. Surko, K. Iwata, C. Kurz, and S. J. Gilbert, \Atomic and Molecular Physics using Positrons in a Penning Trap," Photonic, Electronic and
Atomic Collisions, F. Aumayr and H. P. Winter, eds. (World Scientic,
1997), pp. 383-392.
82
143
70
24
xi
INVITED TALKS
S. J. Gilbert, \Low-energy Positron Scattering from Atoms and Molecules,"
10th Workshop on Low-Energy Positron and Positronium Physics (a satellite conference of the International Conference on the Physics of Electronic
and Atomic Collisions), Tsukuba, Japan (1999).
Fields of Study
Major Field: Physics
Studies in Atomic and Molecular Physics
Professor Cliord M. Surko
Studies in Plasma Physics
Professor Cliord M. Surko
Studies in Positron Physics
Professor Cliord M. Surko
xii
ABSTRACT OF THE DISSERTATION
A New Ultra-Cold Positron Beam and Applications
To Low-Energy Positron Scattering and
Electron-Positron Plasmas
by
Steven Jay Gilbert
Doctor of Philosophy in Physics
University of California, San Diego, 2000
Professor Cliord M. Surko, Chair
A new technique was developed to generate intense, cold, magnetized positron
and electron beams. The beam is formed by extracting particles from a thermalized, room-temperature, single-species plasma conned in a Penning trap. Cold
positrons with an energy spread of 18 meV can be produced either in a pulsed
or continuous mode at energies ranging from < 50 meV upward. Cold, quasisteady-state electron beams have also been generated with electron currents of
0.1 A for several milliseconds using this method. These cold beams have been
used to study both positron-matter interactions and electron-positron plasma
interactions. Positron-atom dierential cross-sections (DCS), positron-molecule
total vibrational excitation cross sections, and total scattering cross sections are
presented. Absolute values of the DCS for elastic scattering from argon and
krypton are measured at energies ranging from 0.4 to 2.0 eV and agree well
with theoretical predictions. The rst low-energy positron-molecule vibrational
excitation cross sections were measured (i.e., for carbon tetrauoride at energies
ranging from 0.2 to 1 eV), and recent extensions of this work to CO, CO2, and
CH4 are described. Total cross section measurements at the lowest positron energies (i.e., down to 50 meV) are also discussed. The electron-positron plasma
study consists of an electron beam transmitted through a positron plasma stored
in a quadrupole Penning trap. The transit-time instability, which is excited by
the beam, was studied from onset through the maximum in growth rate. The
experimental results are compared with the results of a new cold-uid model
and are in good agreement over a broad range of energies and beam currents.
xiii
xiv
Chapter 1
Introduction
Positrons were rst predicted by Dirac [20] in 1930 and discovered soon after
by Anderson in 1932 [3,4]. Anderson made his discovery by studying tracks of
cosmic rays in a cloud chamber. He noticed a particle with a positive charge that
appeared to be lighter than both the proton and the alpha particle (the only
known positive particles at the time). The new particle appeared to have the
same mass as an electron. Anderson called this new particle a positive electron
or positron.
Two years later Joliot generated the rst man-made radioelement, and coincidentally the rst man-made positron source, by bombarding a thin sheet of
aluminum with alpha particles [57]. The isotope 30 P that he created decays into
a stable isotope of silicon by emitting a + (i.e., a positron) and a neutrino.
Thus, Joliot developed the rst radioactive source of positrons. Advances in
positron sources progressed quickly after Joliot's discovery, generating stronger
sources with longer life times.
Since the discovery of the positron [3], it has been clear that the study of
positron interactions with matter is an important and insightful area of physics
research. In order to study positron interactions with matter experimentally, two
criteria must be met. The rst is that the signal-to-noise ratio of the experiment
must be suÆciently large. The second is that the energy and or spatial resolution
of the positrons must be small enough for these results to be meaningful.
The signal-to-noise ratio in a positron-matter experiment is typically related
to the abundance of low-energy positrons and the eÆciency of their detection.
Positrons can either be detected using a scintillator, which detects the -ray
emitted when the positron annihilates with an electron, or by measuring the
positron charge using a charge multiplier. In either case, low-noise positron
detectors with near unity eÆciency are more-or-less conveniently available.
1
2
1.1
Chapter 1
Positron Sources
Positron sources of suÆcient yield to perform experiments have been available
almost since the discovery of the positron. Currently, there are two methods
for producing a high intensity positron source. Positrons can either be created using a linear accelerator (LINAC) [1,48,101] or a radioactive isotope. A
LINAC produces positrons by bombarding a high-Z material such as tantalum
with high-energy electrons ( 100 MeV). The rapid deceleration of the electron generates Bremsstrahlung rays, which in turn create electron-positron
pairs. The positrons are then extracted and formed into a positron beam. The
main advantage of a LINAC-based positron source is that the intensity can be
very high (i.e., > 1011 positrons/s). Disadvantages of using a LINAC-based
positron source include an electrically noisy environment generated during the
high-energy electron pulse, limited availability of LINAC facilities, and relatively
high capital and operating costs.
Radioactive positron sources have improved considerably since Joliot's discovery of a phosphor source with a three minute half-life. The most common
sources in use today are 22Na, 68 Ge, and 58 Co. The main advantages of using
a radioactive source are that they are relatively inexpensive (e.g., compared to
a LINAC source), small, and can be self contained. A self contained source is
critical for the accumulation and trapping of positrons used in the experiments
described in this thesis. The positron accumulation eÆciency is very sensitive to
impurities, such as hydrocarbons, and so having to connect the vacuum chamber to another vacuum system, such as a LINAC, would make maintaining a
hydrocarbon free system more diÆcult.
A 22 Na source is used in the experiments described in this thesis. It has
a 2.6 year half-life and is attainable with activities up to 150 mCi. The only
disadvantage to using a radioactive source over a LINAC based positron source
is a lower positron intensity ( 109 positrons/s for 150 mCi 22 Na source vs.
1011 for a LINAC). For our experiments, the advantages clearly outweigh the
disadvantages making a radioactive source an attractive choice.
Positrons emitted from either radioactive sources or particle accelerators have
a broad energy spread ranging up to several hundred keV and therefore need
to be slowed down before they can be eectively used in a positron-matter experiment. Unfortunately, advances in producing mono-energetic positrons were
not as forthcoming as advances in positron sources. In fact, as late as 1969 in a
review on the theory of `Positron Collisions', Bransden lamented that:
the points of contact between the experiments and theory are not
as many as could be wished and are somewhat indirect. The reason for this is that, in contrast to the electron in positron scattering, there are at present no controlled mono-energetic beams of lowenergy positrons [8].
Introduction
1.2
3
Moderators to Produce Low-Energy Positrons
In 1972, Costello et al. made a critical breakthrough, nding evidence that fast
positrons impinging on a gold surface are slowed to a few electron volts and then
ejected from the surface [17]. This resulted in a practical technique to produce a
low-energy positron beam. Not long after this, improvements on the gold moderator were made by Coleman et al. [15] and then further improved by Canter
et al. [11]. These discoveries led to the rst low-energy positron beams used in
positron-matter experiments, which were measurements of the total cross sections for low-energy positron-helium collisions [11]. Canter's moderator consisted
of a system of gold vanes. Each vane had a ne MgO powder deposited onto it,
increasing the moderating eÆciency by a factor of 10 from that of the gold vanes
alone. A positron beam of variable beam energy was produced by extracting the
moderated positrons through an electrostatic eld. The nal moderator had an
energy spread of 1.5 eV FWHM and a yield of 10 5 , where is the ratio of
low energy positrons extracted from the moderator to the high energy positrons
impinging on the moderator.
Work on producing more eÆcient positron moderators has progressed continually since Canter's work [11]. Single crystal metal moderators, such as tungsten [33], nickel [110] and copper [77] have eÆciencies as large as 10 4 .
Tungsten is especially appealing because of its narrow energy distribution, 0.3 eV
FWHM, making possible some of the highest resolution positron-matter experiments to date [60, 100]. The introduction of solid rare-gas moderators in the
1980s improved positron moderator eÆciencies by another two orders of magnitude [80]. The most eÆcient rare-gas moderator to date is made by freezing
neon gas onto a metal surface at 8 K. For the experiments described in this
dissertation, a solid neon moderator was used. It has an eÆciency 10 2 , and
an energy spread of 1 eV FWHM.
Until the work presented in this thesis, almost all eorts to reduce the energy
spread in the available positron beam sources and to increasing positron brightness have been focused on improved moderator schemes. One successful method
is through the use of remoderators. It has been shown that positron moderator
eÆciency increases as the energy of the incident positron decreases, and can be
as large as 0:3 for a 3 keV incident positron [9]. There is also evidence that
the energy distribution of the emitted positrons is nearer to that of a thermalized
distribution at the moderator temperature. For example, the moderated beam
energy distribution of a 3 keV incident positron beam on a Ni(100) moderator at
300 K and 23 K is 80 meV and 24 meV FWHM, respectively [25]. Unfortunately
the moderation eÆciencies are greatly reduced as the moderator temperature is
decreased [9], and so a cold beam can only be generated using this technique at
the expense of a low overall moderator eÆciency.
Another use of remoderators is to increase the positron beam brightness (i.e.,
ux per unit area per unit energy). This can be accomplished by accelerating
4
Chapter 1
and then electrostatically focusing a moderated beam onto a second moderator.
The re-emitted positrons will have a spatial resolution comparable to that of
the focused beam and an energy resolution 0:1 eV. By successive acceleration,
focusing, and remoderation steps, smaller beam sizes can be achieved without
the expense of increased energy spread [59]. Because of the high remoderation
eÆciency, the brightness of such a beam can be increased although, as mentioned
above, the positron ux decreases with each remoderation step.
Since Canter's positron-helium total cross section measurements [11], a large
body of work has been performed to study positron-matter interactions at lowenergies. Examples in atomic physics include total cross section and positron
annihilation rate measurements, inelastic cross sections for positronium formation, excitation and ionization of atoms, and dierential elastic scattering cross
sections. Excellent reviews on positron-atom and positron molecule cross sections measurements can be found in Refs. [12,53,60]. An account of the earlier
work on measurements of total cross sections can be found in Refs. [40,73]. There
have also been great advances in the study of surfaces using slow positron beams,
including defect-depth proling, low energy positron diraction and reection,
and high-energy positron diraction [78,94].
1.3
Positron Physics at Lower Energies
Despite these developments, there is still a largely unexplored region of energy
(i.e. < 1 eV) which cannot be studied easily using the existing moderated beams.
This is a very interesting energy regime. For example, in atomic physics many
important processes occur at these low energies, such as vibrational [19,28,62]
and rotational [30] excitation of molecules, which have not yet been experimentally studied by positron impact. The low-energy interaction of positrons
and ordinary matter is important in elds such as astrophysics, atomic physics,
and chemical physics. Exploring this energy regime should provide important
new information, such as understanding the role of virtual positronium states in
positron interactions with matter [53], the mechanisms by which positrons bind
to atoms and molecules [39], and the process of large molecule fragmentation by
positrons [47,89].
1.4
Overview of the Thesis
This thesis describes a new technique to produce a state-of-the-art mono-energetic
positron beam [32,65]. To form a cold beam, positrons are rst accumulated in
a Penning trap where they thermalize to room temperature through collisions
with a background gas. These cold positrons are then extracted from the trap by
decreasing the depth of the potential well conning them, thus forcing the cold
positrons out of the potential well and into a beam. The beam has an energy
Introduction
5
distribution of 18 meV FWHM, and it can be tuned over a wide range of energies
from < 50 meV upward. Both pulsed and steady-state beams can be produced
depending on the requirements of the experiment at hand. Positron throughput
is > 1 106 positrons/s, and in pulsed operation, the beam brightness is greater
than that achieved using two remoderation stages [94].
This thesis also describes the rst uses of this new beam in two areas of
positron-matter interactions. The rst is a study of positron-atomic and positronmolecular physics at energies below 1 eV. We have measured the dierential cross
sections for positron collisions with argon and krypton at energies below that of
any previous measurement. We have also made the rst measurements of the
cross sections for vibrational excitation of molecules by positrons, studying the
excitation of CF4 at positron energies as low as 0.2 eV. Most recently we have
extended our study of positron-molecular vibrational cross sections to include
CO, CO2, and CH4 . We have also recently studied the total cross section for
positron-molecule scattering at energies from 50 meV to several electron volts,
which represents a higher energy resolution measurement than any previous
work. Both the technique developed to produce the cold positron beam and the
new method to measure scattering cross sections are in the early stages of development. We expect improvements in both will continue to extend our ability
to explore atomic and molecular physical processes at energies below 1 eV.
The second area of positron-matter interactions we have examined using the
cold beam is the study of electron-positron plasmas. Because of diÆculties in
simultaneously conning both positrons and electrons, the simplest experimental
arrangement in which electron-positron plasma interactions can be studied is
an electron beam passing through a positron plasma. The unique ability to
accumulate and store large numbers of positrons that the group has developed
greatly facilitates this kind of experiment. The work by Greaves et al. [34]
was the rst experimental study of this system; it was done by transmitting
an electron beam through positron plasmas stored in Penning traps with both
cylindrical and quadrupole potential wells. In these experiments, a conventional
hot-cathode electron gun was used as the electron beam source. Unfortunately,
the large energy spread of the hot-cathode electron gun restricted beam-plasma
studies to energies above 1 eV.
Although conventional means to create mono-energetic electron beams are
available, they do not work in the high magnetic eld ( 1 kG) necessary to
conne the positron plasma. We have been able to apply the same technique used
to produce cold positron beams to generate cold electron beams. Because this
technique was designed to operate in the high eld needed for the beam-plasma
experiments, it was an ideal way to carry out the beam-plasma experiments
at the lower beam energies where the maximum growth rate and instability
onset were predicted to occur. This thesis discusses research using the cold
electron beam to investigate further the instability generated by passing a cold
electron beam through a positron plasma conned in a quadrupole well. We
6
Chapter 1
were able to study the instability down to beam energies as low as 0:2 eV,
which corresponds to the onset of the instability.
The organization of the thesis is as follows. Chapter 2 discusses specic
details of the experimental apparatus which pertain to all of the experiments
described in this thesis. The technique used to generate cold positron and electron beams is presented in Chapter 3 along with specic characteristics of the
types of beams. Chapters 4 and 5 describe new experiments using the cold beams
to study positron-matter interactions. In Chapter 4 positron-atom and positronmolecule studies are described in the largely unexplored range of energies below
1 eV. Positron-electron plasma physics studies are described in Chapter 5, in
the form of an electron-beam positron-plasma transit-time instability. Finally,
a summary of the work and concluding remarks are presented in Chapter 6.
Chapter 2
General Description of the
Experiment
This chapter describes the apparatus and techniques used which are common to
all aspects of the experiments discussed in this thesis. Positrons emitted from a
radioactive 22Na source are moderated to low energies and magnetically guided
into a three stage Penning trap used to accumulate the positrons. The entire
system is enclosed in a ultra-high vacuum (UHV) chamber, which has achieved
pressures as low as 6 10 11 torr. A conning magnetic eld is generated using
a number of solenoids which surround the vacuum vessel. The eld varies from
200 G in the source chamber and beam tube up to 1500 G in the accumulator,
where a high eld is necessary for good positron connement. Once a plasma is
accumulated it is either used as a reservoir to form a cold beam as described in
Chapter 3, or as a cold plasma in a beam-plasma experiment (see Chapter 5).
The apparatus described in this chapter has been completely redesigned from
the knowledge gained by operating an earlier version of a positron accumulator.
A detailed description of the earlier accumulator can be found in Ref. [84]. Some
of the work described in this thesis was done using the earlier apparatus. In these
cases, a note will be made in the text that the earlier accumulator was used,
and any additional information pertaining to the particular experiment is given.
However, in most cases, the operation of these two machines is similar enough
to not warrant this.
2.1
Charged Particle Motion in a Magnetic Field
Because all of the experiments described in this thesis are conducted in a magnetic eld, it is helpful to briey review charged particle motion in such a eld.
The most basic motion is that of a charged particle moving through a constant magnetic eld. Figure 2.1 shows a schematic diagram of this motion. The
particle follows a helical orbit which can be conveniently split into two distinct
7
8
Chapter 2
Figure 2.1: Charged particle motion in a magnetic eld. The helical path can be
separated into a circular motion in the plane perpendicular to the eld and a linear
motion along the eld. The total kinetic energy, E , of the particle is the sum of the
kinetic energy associated with motion along the eld, Ek, and the kinetic energy due to
the circular motion, E?, where E = E? + Ek.
motions, a linear motion along the magnetic eld and a circular motion in the
direction perpendicular to the eld. The radius of this orbit, known as the
cyclotron radius rc, is proportional to the particle velocity and inversely proportional to the magnetic eld strength. Specically, rc = mv?=eB , where m, is the
charged particle mass, v? is the particle velocity perpendicular to the magnetic
eld, e is the charge, and B is the magnetic eld strength. For example,pa typical
positron in our cold positron beam (see Section 3.3) will have v? 2kT=m,
where kT is the thermal energy of the room temperature positrons ( 0:025 eV).
The cyclotron radius of such a positron when placed in a 0.1 tesla eld will therefore be, rc 5 m, which has been greatly exaggerated in Fig. 2.1 to show the
helical motion.
It is convenient to split the total particle kinetic energy into the components
due to these two motions. We express the total energy of the particle as E =
E? + Ek , where Ek is the kinetic energy along the magnetic eld and E? is
the kinetic energy in the circular motion perpendicular to the magnetic eld.
Chapter 4 describes how this convention simplies the analysis of the scattering
events in the strong magnetic eld.
When the magnetic eld strength varies as a function of position, the trajectory of the charged particles can be described with the help of a useful adiabatic
invariant. The ratio E?=B is adiabatically invariant as long as the distance
over which the magnetic eld strength changes appreciably is small compared
to the cyclotron radius. For example, as a charged particle moves into a region
of decreasing magnetic eld E? must decrease. Conservation of energy implies
that Ek must therefore increase. As the particle continues to move into a weaker
eld, Ek increases until nearly all of the particles energy is in Ek . At this point
the charged particle is moving almost directly along the magnetic eld with
a very small cyclotron radius. Section 4.3.2 describes how we take advantage
of the adiabatic invariant to determine the total vibrational cross section of a
positron-molecule scattering event.
General Description of the Experiment
9
Lastly, when the lines of magnetic induction curve, and the radius of curvature R is large compared to the cyclotron radius, the zero-order approximation
to the motion of the particle in the eld is to follow the lines of force. For the
experiments described here this approximation always holds, and therefore the
guiding center of the particles, to zero-order, always follows the magnetic eld.
To rst-order there is a drift velocity in the guiding center motion, associated
with the curvature R. This motion, which is in a direction perpendicular to
the magnetic eld, is too small to eect the particle trajectory in a single pass,
for example, when transferring the positrons from the source to the accumulator (see Section 2.3). When the particles make many passes through a curved
region of eld the accumulative drift can be quite large. For this reason, the
magnetic eld is highly uniform throughout the positron accumulator, therefore
minimizing any drifts which would reduce the connement time of the trapped
positrons.
2.2
Source and Moderator
There are several possible approaches to generating the slow positron beams
necessary for trapping and accumulation [88,94]. In all of these approaches the
positrons originate from either a radioactive source or from a particle accelerator.
In the case of radioactive sources, one of the most intense positron producers
is 64 Cu. Although the fast-positron count rate for 64 Cu is quite large (1 1012 e+ =s), the production requires a reactor with a high thermal neutron ux
and must be generated daily because it has a half-life of only 12.8 h [69]. For these
reasons, we have chosen to use the more practical radioactive positron source
22 Na. 22 Na has many properties which make it a good choice, most importantly
it has a high branching ratio of 90%, a long life time, and is commercially
available.
The 22Na positron source that is used in the experiment was obtained from
DuPont Merck Pharmaceuticals in September 1997 with a source strength of
150 mCi and a quoted eÆciency of 70% of the 2 value. The 22Na source has
a half life of 2.6 y and emits a broad energy range of positrons with a fairly
continuous spectrum up to 540 keV. The source is sealed in a titanium capsule
used to isolate it from the vacuum system. The current source eÆciency is
reported to be a factor of two higher than past eÆciencies. This improvement is
obtained by increasing the purity of the source material, and therefore reducing
positron annihilation.
In order to slow the keV positrons to eV energies a moderator is needed
[13,41,79,80,94]. Positron moderators take advantage of positron interactions
with solids as follows. A high energy positron hits the solid and initially looses
energy by ionization or creation of electron-hole pairs. At lower energies, the
positron loses energy by positron-phonon interactions and eventually thermalizes
10
Chapter 2
Figure 2.2: Schematic diagram of the positron source and moderator showing the
positron source relative to the moderator cone. The cone is kept at 8 K by a two-stage
refrigerator, which is in thermal contact through an elkonite rod with the cone. A heat
shield surrounds the source and cone to reduce the radiative heat loss.
with the solid. Because the time it takes for the positron to thermalize ( 10 12 s)
[64] is short compared to their annihilation lifetime, a fraction of the thermalized
positrons can diuse (via positron-phonon collisions) to the surface. By using a
solid with a positive work function, a portion of these positrons are then ejected
from the solid with an energy comparable to the positron work function of the
solid.
There are a number of dierent types of moderators in use. Single-crystal
metal moderators, such as tungsten [33], nickel [110] and copper [77] were originally used, and have eÆciencies as large as 10 4 . More recently rare-gas
solid moderators such as neon have been shown to have much higher eÆciencies ( 2:6 10 2 ) [35], and are therefore used in these experiments. The
minor draw-back of a larger energy spread ( 1 eV FWHM in the neon moderator vs. 0:3 eV FWHM for the tungsten) is not an important factor because
the positron accumulator (see Section 2.4) is almost as eÆcient at trapping a
positron beam generated from the neon moderator as it is from the tungsten
moderator [52].
Figure 2.2 is a schematic diagram of the source and moderator arrangement.
The 22Na source is located in a titanium capsule which has a 13 m titanium
window welded onto its front. The titanium capsule is attached to an elkonite
rod (tungsten-copper alloy used for its high thermal conductivity and good -ray
shielding abilities), which is, in turn, attached to a two-stage refrigerator. An
General Description of the Experiment
11
Figure 2.3: Schematic diagram of the positron source and moderator vacuum chamber
showing the source and moderator mounted to the two-stage refrigerator. The source
assembly is oset from the beam tube to block the line of sight from the source to
the accumulator. A magnetic eld generated by a set of pancake coils and a vertical
coil guides the moderated positrons into the beam tube. Radiation from the source is
blocked by lead bricks, which surround the vacuum chamber.
OFHC aluminum cone is screwed on over the titanium capsule and serves as a
cold surface to form a frozen neon moderator. The cone is in thermal contact
with the second stage of the refrigerator and is typically cooled to 8 K, which
is measured using a calibrated silicon diode. Thermal regulation is obtained by
a heating coil located on the refrigerator and a control feedback system. To
reduce the heat load on the second stage, a heat shield (35 K) surrounds the
entire source/moderator assembly and is in thermal contact with the rst stage
of the refrigerator, which has a greater heat capacity.
A schematic diagram of the UHV vacuum chamber that houses the source and
moderator is shown in Fig. 2.3. The vacuum system is pumped by a ion pump
and has a typical base pressure of 5 10 8 torr. The positron moderator
is generated by introducing neon gas directly in front of the 8 K moderator
cone (see Fig. 2.2). We have no direct measurement of the neon gas pressure
near the moderator cone, and instead regulate the pressure using a stable ion
gauge located in the UHV chamber. The pressure outside is maintained at
2 10 4 torr, and one can assume that the pressure near the cone is much
higher. The moderated positron count rate is monitored as a function of time
during the grow cycle by a small -ray detector located near the beam tube,
and the growth cycle is stopped when the positron count rate saturates, which
typically takes 1 h. After the moderator has nished growing, its eÆciency
continues to rise for 1 h, perhaps by a rearrangement of the neon crystal
12
Chapter 2
structure. A moderator grown in this fashion typically yields a positron ux
of 6 106 e+/s and can last for many months. The moderator used in our
earlier apparatus had a life time of 12 h, making the calibration of long term
experiments diÆcult. The current improvement, which we attribute to a cleaner
UHV system, eliminates this diÆculty. A detailed description of the solid neon
moderator apparatus and operation can be found in Ref. [35]
2.3
Beam Extraction and Transport
The neon moderator is biased to 30 V for beam extraction and eÆcient
positron accumulation (see Section 2.4). The moderated positrons are guided
from the moderator to the accumulator by a magnetic eld ( 200 G) generated
from the series of pancake coils shown in Fig. 2.3. Pancake coils are used so
that lead shielding can be located as close to the 22 Na source as possible. To
prevent the 1.27 MeV -rays and high-energy positrons emitted from the source
from interfering with the -ray detectors located beyond the accumulator, the
source and moderator are oset from the axis of the positron accumulator by
2 cm. A vertical coil is wound around the source chamber to transport the lowenergy positrons from the oset position onto the axis of the accumulator, while
the high energy positrons and -rays hit the wall of the source chamber. The
positrons then enter the beam tube, which is a small diameter ( 2 cm) vacuum
tube surrounded by a magnetic coil. They are then guided into the positron
accumulator.
2.4
Positron Accumulation
The low-energy positron beam enters the positron accumulator at a rate of
6 106 e+/s. A specially designed Penning-Malmberg trap [38,84,104] is then
used to eÆciently trap the positrons. Figure 2.4 shows a schematic diagram of
the modied trap used in the experiment. The trap uses a set of cylindrically
symmetric electrodes to produce an electrostatic potential well that connes the
positrons axially. Radial connement is achieved with a magnetic eld generated
along the axis of the electrode structure. This design, which oers excellent longterm connement of positrons [71,86], enabled the creation of the rst laboratory
positron plasma in 1989 [104].
Trapping the slow positron beam as it enters the accumulator is a nontrivial task. For the case of an abundant charged particle, such as electrons, an
acceptable trapping scheme is simply to raise a potential barrier when the trap is
ooded with particles, thereby trapping all of the particles within the electrodes.
Because electron beams with densities nb > 109 cm 3 are easily attained, electron
plasmas with the same density can be trapped using the above technique. For
the case of positrons from our 22Na source and moderator the beam density is
General Description of the Experiment
13
Figure 2.4: Schematic diagram of the three-stage positron accumulator, showing the
electrode structure (above), which is used to create regions with dierent pressures of
nitrogen buer gas by dierential pumping. (below) The electrostatic potential prole
used to trap the positrons.
only nb > 10 1 cm 3, therefore an eÆcient means of accumulating positrons over
a relatively long period of time must be used.
To achieve this, a buer-gas trapping scheme is used in the following manner.
The moderated positrons enter the positron accumulator with a beam energy of
32 eV and an energy spread of 1 eV FWHM. A nitrogen buer gas is introduced into the middle of the rst stage of the accumulator and is pumped out at
both ends. Dierential pumping is used to generate three pressure regimes from
10 3 down to 10 6 torr, corresponding to the three stages. As the moderated
positrons enter the rst stage of the accumulator they inelastically collide with
the N2 buer gas (\A" in Fig. 2.4) losing energy, and becoming trapped in the
potential well. By making another two inelastic collisions with the buer gas
(\B" and \C") the positrons move from the relatively high pressure region of
stage I into the low pressure region of stage III, where they cool to room temperature (0.025 eV) in approximately 1 s by further collisions with the N2 buer
gas [38].
The most eective inelastic process for trapping the positrons is electronic
excitation of N2 by positron collision at approximately 8.6 eV. In order to max-
14
Chapter 2
imize the trapping eÆciency via the electronic excitation, the potential well
depth in each of the stages must be adjusted so that the cross section of the
electronic excitation is maximized through that stage. Unfortunately, another
dominant cross section which turns on at energies near the electronic excitation
is positronium formation at 8.8 eV. Because positronium formation is a positron
loss mechanism it is critical to operate the accumulator at energies below where
this is a dominant loss mechanism. In practice, the maximum trapping eÆciency is found by mapping out the trapping rate as a function of well depth
and searching for a maximum. Typically the maximum eÆciency occurs when
the accumulator is operated with a step height between stages of V 9 V.
Another critical factor which eects the trapping eÆciency of the accumulator is the buer gas pressure prole through the three stages. In order to trap
the moderated positrons the pressure in the rst stage is adjusted so that the
probability of an inelastic collision occurring on the rst pass through the accumulator is large (labeled \A" in Fig. 2.4). Once the positron has been trapped it
can make multiple passes through all three stages until the inelastic collision labeled \B" occurs and the positron falls into the next potential well. The pressure
in stage II has to be large enough for the second transition to occur before the
positron annihilates on the N2 buer gas in stage I. Similarly, once the positron
is trapped in stages II and III, the transition into the stage III (\C") must occur
before the positron annihilates in stage II.
The pressure of the nitrogen buer gas in all three stages can be adjusted
as follows. The stage I pressure is controlled by adjusting the rate of nitrogen
introduced into the center of the stage I electrode structure. The right end of
the stage I electrode has a 5 cm long slotted section which allows some of the
buer gas to exit the stage I electrode before entering stage II. A bae, which is
externally adjustable, can slide over this slotted region restricting this ow of gas
through the slots (see Fig. 2.4). By moving this bae from a completely closed
to open position the pressure ratio between the stage I and stage II electrodes
can be adjusted from 2 to 20, respectively. The stage III pressure can be altered
by adjusting the ow of a second nitrogen gas line which is located near the third
stage. In practice, the trapping eÆciency of the accumulator is maximized by
adjusting the large parameter space, which includes the accumulator electrode
potentials and the pressure prole through each of the three stages, and searching
for maxima. The electrode optimization is achieved using a computer assisted
optimization routine [38]. The pressure optimization, which is not yet under
computer control, was maximized by manually sweeping the parameter space.
There are two regimes in which the accumulator is operated. Typically the
accumulator is run at its highest trapping eÆciency. Figure 2.4 shows the pressures used in the three stages to achieve the maximum trapping eÆciency, which
for our new accumulator is 20%. Operating the accumulator for maximum
eÆciency requires a fairly high pressure in the third stage and therefore the
positron life time is only 40 s. If a longer positron lifetime is needed the nitro-
General Description of the Experiment
15
10-6
Pressure (Torr)
10-7
Valve shut
10-8
10-9
10-10
0
10
20
30
40
50
60
Time (sec)
Valve open
Figure 2.5: Buer gas pressure in the third stage as the gas is cycled from its base
pressure to its operating pressure and back to its base pressure. The pressure in the
third stage can be decreased by three orders of magnitude in 10 s.
gen buer gas can be quickly pumped. Figure 2.5 shows the pressure in the third
stage of the accumulator as the buer gas is cycled from its operating pressure
of 3 10 7 torr to its base pressure of < 1 10 9 torr in a few seconds. With
the buer gas pumped out the positron lifetime increases to 20 min, limited by
impurities in the vacuum chamber. In the earlier accumulator a dewar lled with
liquid nitrogen, which pumped out these impurities, surrounded a quadrupole
Penning trap, and in this trap positron lifetimes of over 2 hours were possible.
Currently we are constructing a high-eld (5 tesla) positron storage stage, which
will incorporate a 4 K cold trap around the electrode structure, with expected
positron lifetimes of many hours to days (see Section 6.2.1).
When a large number of accumulated positrons are needed it is more eÆcient
to operate the accumulator with a lower pressure in the third stage. Figure 2.6
shows the results of a computer particle code used to simulate the pressure
prole throughout the accumulator [5] when it is optimized for a maximum
number of trapped positrons. Operating under these conditions allows the trap
to accumulate positrons for over 6 min, trapping approximately 3 108 positrons
16
Chapter 2
10-2
I
III
II
pressure (Torr)
10-3
10-4
10-5
10-6
10-7
10-8
0
1
2
3
z(m)
Figure 2.6: Calculated pressure prole as a function of distance along the three-stage
positron accumulator. The pressure prole shown here is used to maximize the total
number of positrons accumulated.
[37].
A schematic diagram of the accumulator and its surrounding vacuum vessel
is shown in Fig. 2.7. The accumulator is contained in a UHV vacuum chamber, bakeable to 130Æ C, with a base pressure which has reached as low as
6 10 11 torr. Two cryo-pumps located at both ends of the accumulator are
used to generate the dierential pumping of the buer gas and to quickly pump
out the gas when needed. The simplied design of the new electrode structure
allows a single magnet to surround the entire vessel (the earlier design had two
magnets), which improves the uniformity of the magnetic eld throughout the
accumulator, and simplies the magnet alignment procedure. The accumulator
is typically operated with a magnetic eld of 1500 G.
2.5
Retarding Potential Analyzer (RPA)
The positron and electron beam energy distributions are measured using a retarding potential analyzer (RPA). An RPA consists of an electrode at a potential
of V0 that rejects all particles with a parallel energy less than eV0 . The particles
General Description of the Experiment
17
Figure 2.7: Schematic diagram showing the three-stage accumulator and surrounding
vacuum chamber.
are guided magnetically through the RPA and those that are not rejected by
the potential barrier are detected on the other side of the analyzer. The RPA
electrode used in these experiments consists of a gold plated aluminum cylinder
with a large length to diameter ratio. This insures that the potential in the center of the electrode is the same as the applied potential. Electrons are collected
on an aluminum plate located beyond the RPA. A 3 k
shunt resistor is used in
conjunction with a voltage amplier to measure the beam current. Positrons are
detected using a NaI(TI) -ray detector, which measures -rays emitted when a
positron annihilates on the aluminum plate.
By measuring the number of particles that pass through the RPA as a function of applied potential, the parallel energy distribution of the beam can be
measured. It is important to understand that only the particle velocity along
the magnetic eld, vk, is aected by the RPA and therefore, only the parallel
energy, Ek is measured. Figure 2.8 shows the parallel energy distribution of a
moderated positron beam measured using the RPA. The lled circles are the
measured data. For applied voltages less than the beam energy (in this case
32 eV) all of the positrons pass through the RPA and are detected. When the
applied voltage is above the beam energy none of the positrons are transmitted through the RPA. As the applied voltage on the RPA scans through the
beam energy from low to high, the coldest positrons are rejected rst, followed
by positrons with larger parallel energies until all of the positrons are rejected.
The open circles in Fig. 2.8 is the derivative of the data showing the FWHM of
1 eV.
number of positrons (arb. units)
18
Chapter 2
1
1 eV
0
28 29 30 31 32 33 34 35 36 37 38
bias voltage (V)
Figure 2.8: Energy distribution of a positron beam emitted from a neon moderator.
Filled circles are measured data. The open circles, which represent the energy distribution, are calculated by taking the derivative of the data.
Chapter 3
Bright, Cold Charged Particle
Beams
3.1
Introduction
This chapter describes a new and versatile method to create bright, cold positron
and electron beams. The beams can be generated in either a quasi steadystate or pulsed mode. There are numerous potential applications for such bright
beams. Examples include material surface characterization, such as defect depth
proling, positron and positronium scattering from atoms and molecules, and
annihilation studies [37,60,78,94]. Furthermore, many applications of positron
beams, such as time-of-ight measurements, positron lifetime experiments, and
time tagging, require pulses of positrons. One advantage of pulsed, as compared
with steady-state beams, lies in the potential for greatly enhanced signal-tonoise ratios in a variety of applications. While several techniques to create
pulsed positron beams have been discussed previously [18,81,105], many of these
techniques have disadvantages. One example is achieving pulse compression at
the expense of degradation in the beam energy spread.
There are several possible approaches to generate slow positron beams [88,
94]. The positrons originate from either a radioactive source or from a particle
accelerator, but in either case they must be slowed from initial energies of several
hundred keV to energies in the electron Volt range before beam formation and
handling becomes practical. At present, this is accomplished most eectively
using a solid-state moderating material (see Section 2.2). In general, positrons
emerge from the moderator with an energy of several electron Volts and an
energy spread in the range 0.3{2 eV, although methods have been described to
reduce this energy spread by as much as an order of magnitude. [10,25].
19
20
Chapter 3
Figure 3.1: Schematic diagram of the experiment. The electrodes are shown schematically in the upper diagram. Below, the solid line represents the potentials applied to
the electrodes: (a) entrance gate, (b) dump electrode, (c) exit gate, (d) RPA, (e) accelerating grids and phosphor screen, (f) NaI gamma-ray detector and (g) CCD camera.
The energies eV0 and eV1 are the electrostatic potentials of the exit gate electrode and
the energy analyzer, respectively.
3.2
Experimental Setup
The upper panel of Fig. 3.1 shows a schematic diagram of the apparatus to
generate and analyze cold beams. Positrons or electrons are accumulated and
cooled in the third stage of the three stage accumulator shown as electrodes
(\a"), (\b") and (\c") in Fig. 3.1. The lower panel of Fig. 3.1 shows the potential
prole of a conning well generated by the electrodes used to store the positron
plasma. A cold positron beam is generated by reducing the size of the conning
well, forcing the positrons over the exit gate electrode. To generate an electron
plasma the same technique is used with the inverse potentials applied to all of
the electrodes.
The radial density variations and temporal evolution of the beam prole
are analyzed using a CCD camera, viewing a phosphor screen located behind
the energy analyzer (\e" and \g" in Fig. 3.1). Particles are accelerated to 8 keV
before being imaged on the screen. For positrons, a 3-inch NaI(Tl) -ray detector
provides measurements of the particle ux. The electron ux is suÆciently
large to be measured by a standard current-to-voltage amplier connected to
Bright, Cold Charged Particle Beams
21
a collection plate. The RPA labeled (\d") is used to determine the energy
distribution of both the electron and positron beams. A detailed description of
the positron accumulation techniques along with an explanation of the RPA can
be found in Chapter 2.
3.3
Positron Beams
For the experiments described here, positrons were accumulated for 10 s, resulting in about 107 positrons in the trap. Because the ll time is much shorter than
the 40-s positron lifetime, the loss of captured positrons during the lling phase
is small. The positron beam is formed within a few milliseconds, so that in this
mode of operation, the duty cycle for accumulation is close to unity. The average
throughput is the same as the accumulation rate of about 1 106 positrons per
second.
A pulsed positron beam is formed using the following procedure. After a
positron plasma has been accumulated in the trap, incremental voltage steps
are applied to the dump electrode (labeled \b" in Fig. 3.1), with each increase
in voltage ejecting a fraction of the stored positrons. After each pulse the dump
electrode is lowered, conning the plasma until the next pulse is formed. During
this process, the entrance gate (\a") is placed 1 V above the exit gate (\c")
to insure that the positrons leave the trap via the exit gate. The energy of
the positron pulses is set by the potential of the exit gate electrode. In order
to achieve a narrow energy spread, it is important that the steps in the dump
voltage are small compared to the plasma space charge, otherwise collective
modes can be excited in the charge cloud, which can, in turn, degrade the
achievable energy resolution [24,44]. Using the central electrode to dump small
fractions of the plasma, the energy of the released positrons is kept the same for
all pulses and is determined solely by the potential of the exit gate.
Figure 3.2(a) shows a pulse train obtained by applying equal-amplitude voltage steps to the dump electrode. For approximately the rst 30% of the pulses
in the train, the pulse height increases and then stays constant for the remainder
of the pulses. The envelope of the pulse train is highly repeatable and unaected
by the number of pulses contained in it. In many cases it is advantageous to
have equal-amplitude pulses throughout the pulse train.
By adjusting the size of the voltage steps in the following manner, it is
possible to compensate for variations in pulse height and achieve a longer attopped region. The integrated charge dumped from the trap for a given pulse
depends only on the dump voltage at the time of the pulse. Because the pulsetrain envelope is highly reproducible, it can be used as an inverse look up table
to determine the voltage steps needed to produce an arbitrary pulse envelope.
The envelope of the pulse train produced with constant height steps in the dump
voltage is shown in Fig. 3.2(a). The results of a wave form adjusted for a constant
22
Chapter 3
arb. units
(a)
arb. units
(b)
0
5
10
15
time (ms)
20
25
Figure 3.2: A pulse train of 60 pulses, each consisting of approximately 105 positrons,
illustrating the ramp-up, at-top and terminal phases of the pulse envelope, which is
independent of the specic number of pulses: (a) corresponds to equal size steps in the
dump voltage, whereas in (b) an adapted step size was used to correct for unequal pulse
amplitudes.
pulse amplitude is shown in Fig. 3.2(b).
To insure that all particles with energies greater than the exit gate potential
have suÆcient time to leave the trap, we apply each voltage step for > 15 s,
which is longer than the axial bounce time in the trap (b 6s). Pulse durations less than b can be produced by increasing the dump electrode potential
for a time shorter than b before returning it to a lower value. This latter protocol produces shorter pulses with a corresponding reduction in the number of
positrons per pulse.
Using a CCD camera, we imaged the radial structure of the pulses, obtaining
a size of about 2 cm (FWHM), roughly equal to the measured plasma size. This
is representative of the rst 75% of the pulse train. Thereafter, the proles
Bright, Cold Charged Particle Beams
23
broaden and become hollow for the last few pulses.
Positron beams with a well-dened energy are important for many applications. Room-temperature plasmas in the trap will equilibrate to an energy
spread corresponding to 12 kT per degree of freedom. We have shown that the
perpendicular energy spread of the plasma is not aected by the dumping process
and remains at the room temperature value [38].
In the regime where the steps in the potential of the dump electrode are
small compared to the plasma space charge, the axial pulse energy spread is not
aected signicantly by the step size or the position of the pulse in the pulse
train. Contributing factors to the axial energy spread include the radial variation
of the exit gate potential across the plasma width, collective plasma eects [44],
and electrical noise on the electrodes. Experimentally, we have shown that the
axial energy spread varies little over pulses in a pulse train.
The axial energy distribution of a pulse taken at the beginning of the attopped portion of the pulse envelope is shown in Fig. 3.3. A description of the
RPA operation is given in Section 2.5. The number of positrons which pass
through the energy analyzer (c.f. Fig. 3.1) is plotted as a function of analyzer
voltage, with the exit gate electrode set at 2 V and a step size of the dump
electrode of 37 mV. An error function is tted to the data and indicates a pulse
centroid energy of 1.69 eV, with an energy spread of 0.018 eV FWHM. We
attribute the dierence of about 0.3 eV between the measured positron energy
and the applied exit gate potential to a combination of contact potentials and
the radial potential gradient in the trap. A lower limit of the beam energy is
expected to be the axial temperature spread of the beam. The highest beam
energy used in this experiment was 9 eV, but this was limited only by the
maximum output voltage of the digital-to-analog voltage converters.
In practice, the pulse repetition frequency is limited at the lower end by
the positron lifetime and at the high end by the positron bounce time. Pulse
amplitudes will be inversely proportional to the number of pulses in the pulse
train. However, if small energy spreads are desired, it is necessary to operate
in the limit where each step in dump voltage is small compared to the plasma
space charge.
We have also created quasi steady-state positron beams of 0.5 s duration with
a current of 0.8 pA. This was done by raising the dump voltage continuously
on a time scale much longer than the particle bounce time. The beam energy
spread in this case was 0.017 eV FWHM, which is comparable to that of the
pulsed positron beam.
We are aware of one other report of a Penning trap used as a pulsed source
of positrons [95]. Positrons from a LINAC, were captured in a Penning trap and
then extracted by applying voltage pulses to the exit gate. However, in this case,
no attempt was made to achieve a well dened beam energy or narrow energy
spread.
Brightness is an important gure of merit for beam sources. Use of the
number of positrons (arb. units)
24
Chapter 3
1
0.018 eV
0
1.5
1.6
1.7
1.8
bias voltage (V)
Figure 3.3: Energy distribution of a pulse. Filled circles are measured data, and the
dashed line is an error function t to the data. The solid line, which represents the
energy distribution, is the derivative of the t.
positron trap and a buer gas to cool the positrons increases their phase space
density, and hence the beam brightness, without signicant loss of beam intensity. The brightness of our pulsed positron beam is 1 109 s 1 rad 2 mm2 eV 1 ,
which is signicantly higher than the brightness reported for a steady state
positron beam with two remoderation stages [94]. In principle, compressing
the stored plasma radially by applying an azimuthally rotating electric eld to
segmented electrodes surrounding the plasma or using a source of positrons at
cryogenic temperatures could enhance the brightness even further (see Section
6.2.1).
Bright, Cold Charged Particle Beams
3.4
25
Electron Beams
A conventional hot-cathode electron gun produces electron beams with an energy spread typically corresponding to several times the cathode temperature
( 0:5 eV). Electron guns optimized for low energy spreads (E 0.2 eV at
a current of several microamps) have been described (e.g., Ref. [58]). However,
their operation in a magnetic eld has not been tested, and the design described
in Ref. [58] cannot be used in a magnetic eld without modications. The energy resolution of an electron beam can be improved by energy lters of various
designs, such as E B lters, spherical deectors, etc. A general discussion of
electron monochromator designs in non-magnetized beams shows that 0.3 A
of beam current presents an upper limit, if the energy uncertainty is to remain
below 0.1 eV [67].
There are other processes which can produce electrons of well dened energies. For example, a synchrotron photo-ionization technique has been described, capable of producing electron beams with an energy uncertainty of
about 3.5 meV. However, the achievable beam currents are low (of the order of
10 12 A) and the processes requires a source of synchrotron radiation.
We have shown that it is possible to generate well dened, accurately controllable, and cold steady-state or pulsed electron beams, by extracting electrons
from a stored plasma, in a manner similar to that described above for positrons.
For example, an electron beam of 0.1 A lasting 4.8 ms can be extracted from
a reservoir of 3 109 cold electrons (i.e. 4:8 10 10 Coulombs).
For the electron experiments, we use a commercial dispenser-type cathode
with a 2.9 mm aperture as an electron source to ll the Penning trap. The
cathode heater current is set so that an extraction voltage of 0.5 V applied to a
grid in front of the cathode results in an emission current of about 2 A. The
simplest method for lling the trap with electrons is to raise a potential barrier
around the trap while the electron source is on (see Section 2.4). This method
cannot be used here because the beam currents attained from the hot-cathode
source are insuÆcient to generate the needed electron densities. Instead the
following trapping technique was used.
Filling the trap with electrons was achieved without the use of a buer gas by
creating a conning well of gradually increasing depth, so that an approximately
constant trapping potential is maintained. The nal well depth of 90 V is reached
in 100 steps in a total time of about 1 s. The resulting electron plasma, which
cools to room temperature by collisions with a neutral background gas [66],
contains 3 109 particles and has a space charge of 90 V. Using a nitrogen
buer gas while accumulating the electrons does increase the trapping eÆciency.
Unfortunately, this also greatly accelerates the radial transport, resulting in an
increased electron beam diameter, which needs to be small for many applications.
After an electron plasma has been accumulated in the trap, a cold beam
is generated by continuously reducing the depth of potential well conning the
26
Chapter 3
20
potential (V)
0
space charge
potential
(full well)
-20
-40
-60
-80
(a)
-100
entrance
gate
dump electrode
exit gate
modification
(b)
0
20
40
60
80
axial distance (cm)
Figure 3.4: (a) The axial potential distribution in the empty trap. (b) A schematic
drawing of the electrode conguration. An electrode added into the electrode structure
(shown in hash) gives rise to the potential depicted by the solid line. The dashed line
was calculated without the modied electrode in place. The approximate position of
the electron plasma is shown in (b).
electrons. As the magnitude of the potential well decreases, the electrons are
forced over the exit-gate electrode, which sets the electron beam energy. The
entrance-gate electrode is set 1 volt below the exit-gate electrode to insure that
the electrons leave via the exit-gate electrode.
The large number of electrons in the trap (3 109 , as compared to 107
positrons) creates a space charge, which can distort the potential near the exit
gate electrode and lead to large uncertainties in the resulting beam energy. The
modied exit gate electrode design shown in Fig. 3.4 increases the separation
between the charge cloud and the exit gate and thereby decouples the extracted
Bright, Cold Charged Particle Beams
27
beam energy from the number of particles stored in the trap. A self-consistent
Poisson-Boltzmann calculation, which includes the modied electrode geometry
in the presence of the electron plasma, conrmed that the plasma potential has
a negligible eect on the resulting beam energy.
The energy distribution of the beam is measured using an RPA (see Section
2.5). Electrons which pass through the RPA are collected on an aluminum plate
and recorded on a digital oscilloscope. The plate is biased slightly positively
( 2 V) to insure that all of the electrons are collected. A potential bias much
larger than this is avoided, since it can cause secondary electron emission from
the aluminum plate, which leads to an apparent increase in the beam current.
Care must be taken while measuring the energy distribution of the electron beam
in a magnetic environment. In particular, when the beam is partially reected
by the analyzer electrode, the reected particles interact with the incident beam,
causing a space charge to build up. This increased space charge can force electrons through the RPA, which appears as an increase in the energy spread of
the beam. To minimize this eect, the retarding energy analyzer is raised for
only a short time ( 10 s) and then lowered to release any charge build up.
The axial energy distribution for an 0.1 A electron beam was measured in
this manner (c.f. Fig. 3.5) and indicates an energy spread of 0.10 eV FWHM.
Both experiment and potential calculation using a Poisson-Boltzmann solver
(c.f. Fig. 3.4) conrm that the absolute beam energy is set accurately by the
externally applied exit gate potential. Larger beam currents have also been
generated and show a corresponding increase in the axial energy spread, which is
believed to be at least in part due to the increased space charge of the beam. For
example, a 1 A electron beam has an energy spread of approximately 0.5 eV. It
should also be possible using the technique described here to generate an electron
beam with an energy spread as low as that of the positron beam (0.018 eV). In
this case the achievable beam currents would be comparable to those used in
the positron beam ( 1 pA). Unfortunately, our present measuring technique
is unable to detect a current this small. Future plans to install a microchannel
plate in the experiment will allow us to explore the possibility of studying these
low-energy electron beams.
As with positrons, having the ability to generate an electron beam with a
fast rise time and constant current thereafter is often advantageous (see Chapter
5 for an example). This can be accomplished by modifying the dump waveform
in the following manner. The total charge dumped from the well at any given
time during the beam extraction depends only on the dump voltage at that
time. Therefore, the time-integrated current (i.e. total charge) resulting from
a beam dump using a linear ramp can be used as an inverse lookup table to
nd the dump voltage required to achieve an arbitrary current waveform. The
current waveform produced by dumping the electrons using a simple linear ramp
is shown in Fig. 3.6(a), and has a rather slow current rise of over 1 ms. By using
this waveform as an inverse lookup table an electron beam with an arbitrary
28
Chapter 3
electron signal (arb. units)
1
0.10 eV
0
1.8
1.9
2.0
2.1
2.2
2.3
2.4
bias voltage (V)
Figure 3.5: Energy distribution of a 0.1 A quasi steady-state electron beam. Filled
circles are measured data, and the dotted line is an error function t to the data. The
solid line, which represents the energy distribution, is the derivative of the t.
waveform can be generated. Figure 3.6(b) shows a beam current generated
using this technique to produce a quick current rise (tr 3 s) and 0:08 A
current thereafter.
3.5
Chapter Summary
We have shown that room-temperature plasmas stored in Penning traps can
be used as versatile sources of pulsed and steady-state beams of positrons and
electrons. In the case of positrons, we measured an energy spread of 0.018 eV
for both pulsed and quasi steady-state beams. Electron beams were extracted
from a plasma of 3 109 particles. At a current of 0.1 A, beams lasting 5 ms
can be formed with an energy spread of 0.10 eV (FWHM). It is likely that the
performance of these cold, bright, electron and positron beam sources can be
further improved by relatively straight forward modications of the techniques
described above.
Bright, Cold Charged Particle Beams
29
0.10
current (µA)
(a)
0.05
0.00
0.10
current (µA)
(b)
0.05
0.00
0.0
0.5
1.0
1.5
time (ms)
Figure 3.6: (a) Current waveform of an electron beam obtained using a linear voltage
ramp to dump the plasma. (b) Current waveform using an optimized dump voltage
waveform to dump the plasma. Note the much faster turn-on ( 3 s) and the extended
region of constant current.
30
Chapter 3
Chapter 4
Positron Scattering from
Atoms and Molecules
4.1
Introduction
The interaction of antimatter with matter is an interesting and active eld of
study [39,43,47,53,54,60,62,68,79,89,94,100,114]. One of the simplest of these
types of interactions is that between a positron and an atom or molecule. Such
interactions are important in atomic physics [39,43,53,54,60,62,68,100,114] and
surface science [46,94], and they have potential technological applications such
as mass spectrometry [47,89]. Although some aspects of these interactions have
been studied in detail [12,54,60,114], most positron scattering experiments to
date have been limited to positron energies greater than 1 eV due to technical
limitations. For example, before the work described here, the lowest energy
positron dierential cross-section d=d
(DCS) measurements were from argon
at 2.2 eV [16], and the only known positron total vibrational excitation cross
sections were for CO2 at energies above 2 eV [62]. The only exceptions to
this are total cross sections which have been measured at energies below 1 eV
[43, 60, 100]. There are, however, many interesting questions at low values of
positron energy, including the existence of positron bound states with atoms or
molecules [91], the role of vibrational excitation in the formation of long lived
positron-molecule resonances [39, 53], and understanding the fragmentation of
large-molecules stimulated by positrons [47,89].
To form a cold positron beam, the limited supply of positrons must be used
eÆciently. Until now, most positron scattering experiments have used moderated
beams of the form discussed in Section 2.2 as a cold positron source, limiting
the experiments to energies greater than about 1 eV [43,60,62,68,100,114]. In
Section 2.4, we describe a novel technique that we developed to overcome these
limitations to achieve a high intensity, cold, magnetized positron beam (i.e.,
18 meV FWHM) [32,65]. Using the cold beam, we were able to make new kinds
31
32
Chapter 4
of positron scattering measurements. These include measuring positron-argon
and positron-krypton DCS at energies lower than any previous measurements
(0.7{2.0 eV), and making the rst low-energy measurement of positron-molecule
total vibrational cross sections at energies as low as 0.2 eV, studying the 3
excitation of CF4 [31].
Aside from some total cross section measurements [43,60,100], positron scattering experiments have been traditionally carried out using moderated positrons
in an electrostatic beam. The positrons are focused onto a highly compact target such as a gas jet, which denes the scattering angle precisely. Typically, a
channeltron detector with a retarding potential grid in front of it is placed on
a movable arm to measure the DCS [49]. Because our cold positron beam is
formed in a magnetic eld, it is expedient for us to conduct the positron scattering experiments in a eld of comparable magnitude. This led us to combine
existing techniques and develop new ones in order to perform measurements
such as elastic DCS, total inelastic cross sections, and total cross sections for
positrons scattering in a magnetic eld.
Many of the techniques described in this chapter have been used elsewhere
in one form or another. For example, positron-atom total cross sections are
typically measured using a gas cell and a positron beam which is guided through
the cell by a weak magnetic eld [43,60,100]. RPA measurements have been used
previously in conjunction with a spatially varying magnetic eld to characterize
the energy distributions of moderated positrons [41]. Positron-atom DCS have
been measured in a magnetic eld using time of ight methods [16]. Magnetized
pulsed beams have been created using a non-thermal reservoir of positrons [61,95,
105]. In addition to combining these techniques in a unique fashion, the ability
to resolve inelastic from elastic scattering events using a varying magnetic eld
between the scattering and analysis regions [see Section 4.3.2], and the use of an
N2 buer gas and three-stage Penning trap to from a cold pulsed or continuous
beam [32, 65, 104] are new developments in the eld of positron atomic and
molecular physics. By combining the cold positron beam formed in this manner
with methods to perform scattering experiments in a magnetic eld, we have
been able to extend the limits of positron-atom and positron-molecule scattering
experiments into a new range of positron energies.
Using the information gained from initial experiments performed on our earlier apparatus, a new scattering apparatus was constructed to be used in conjunction with the new positron accumulator (see Section 2.4) [103]. Initial work
using the new accumulator and scattering apparatus has improved our capability
to study low-energy positron physics, because of the brighter positron beam and
increased positron throughput. The new scattering apparatus also has the capability to operate at a magnetic eld ratio four times greater than that previously
possible. As discussed, this provides correspondingly greater discrimination for
inelastic scattering studies. These improvements lead to increased signal-to-noise
ratio and improved energy resolution. The initial results from the new appa-
Positron Scattering from Atoms and Molecules
33
ratus are promising. The studies described here are the beginning of a broad
experimental program in low energy positron atomic physics.
This chapter is organized in the following manner. First, scattering from
atoms and molecules in a magnetic eld is described, followed by a discussion
of the methods used to measure dierential elastic, inelastic, and total cross
sections. Measurements of elastic positron DCS on argon and krypton, and
inelastic vibrational cross sections for CF4 , are then summarized. The chapter
continues with a discussion of other eects relevant to this kind of measurement,
and concludes with a set of remarks emphasizing the future of experiments in
this area of low-energy positron atomic physics.
4.2
Experimental Setup
The scattering experiments are conducted in the following manner. First, positrons are accumulated for 0.1 s and then cooled to room temperature in
1 s. A cold beam of approximately 105 positrons is formed using the technique
described in Section 3.3. It is guided magnetically through the scattering cell
where it interacts with the test gas. The parallel energy distribution of the
scattered beam is then measured with an RPA and analyzed using the techniques
described in Section 4.3. In order to improve the statistics, the entire 2 second
sequence is typically repeated 1000 times, and so a typical data run takes 11
hours to complete. A schematic diagram of the scattering experiment is displayed
in Fig. 4.1(a), showing the third stage of the positron accumulator, the scattering
cell and the RPA. The scattering apparatus is located in a UHV vacuum chamber
which attaches to the end of the three stage accumulator. Two 60 cm magnets
are used to generate a uniform eld through the scattering cell and RPA.
Test gas is continually introduced into the center of the scattering cell, which
is 38 cm long with a 1 cm internal diameter, and is pumped out at both ends using
two cryo-pumps [see Fig. 4.1(a)]. A well localized region of higher gas pressure is
created by dierential pumping between the 1 cm diameter scattering cell and the
9 cm diameter vacuum chamber. Figure 4.1(b) shows the calculated pressure
prole of the test gas in the scattering apparatus. A typical gas pressure of
0.5 mtorr in the center of the gas cell is reduced by over two orders of magnitude
by the ends of the cell. This permits operation of the experiment in steadystate with the test gas isolated in the scattering cell and the nitrogen buer
gas conned to the accumulator. In our previous scattering apparatus it was
necessary to pump out the nitrogen buer gas before the test gas could be
introduced. By eliminating this complication we have been able to increase the
positron throughput by an order of magnitude.
To further reduce the eects of the test gas on the accumulator ll cycle,
a 1.2 cm diameter gas bae was placed between the scattering cell and accumulator [see Fig. 4.1]. In order to reduce the eects of multiple scattering, the
34
Chapter 4
(b)
pressure (mtorr)
(a)
1
0.1
0.01
0.001
0.0001
Figure 4.1: (a) Schematic diagram of the scattering apparatus showing the third stage of
the positron accumulator, the scattering cell and the RPA, which are located in a UHV
chamber. Magnetic coils labeled 1 through 3 are used to generate a uniform magnetic
eld through the accumulator, scattering cell, and RPA, respectively. (b) Calculated
test-gas pressure prole along the axis of the scattering apparatus. The gas pressure
near the ends of the scattering cell is more than two orders of magnitude smaller than
at the center.
test gas pressure in the scattering cell is adjusted so that 10% of the positron
beam is scattered by the gas. We are able to determine the average absolute
pressure in the scattering cell, with better than 1% absolute accuracy, using a
thermally regulated capacitance manometer gauge, which directly measures the
pressure at the center of the scattering cell. Because capacitance manometers
use a direct pressure measurement (unlike an ion gauge), the pressure reading
is independent of the specic test gas being studied. We verify for the atoms
and molecules studied that the operating pressures used in the scattering cell
(typically 0.1 to 1 mtorr) are in a pressure regime where the measured scattering
cross sections are independent of the test gas pressure.
Positron Scattering from Atoms and Molecules
35
Figure 4.2: Positron scattering in a magnetic eld. A positron from a cold beam, with
most of its kinetic energy in the parallel component, Ek, follows magnetic eld until it
scatters from an atom [see inset]. The scattering event transfers some total energy, E , of
the positron from Ek into E?, depending on the scattering angle and atomic processes
involved. The scattered positron continues along the eld with an increased value of
E? and decreased Ek .
4.3
Data Analysis
The scattering measurements presented here exploit the behavior of positrons
in a 0.1 tesla magnetic eld. As mentioned in Section 2.1, the total positron
energy, E , can be expressed as E = E? + Ek, where E? and Ek are the contributions to the motion perpendicular and parallel to the magnetic eld. Figure
4.2 depicts a positron scattering from an atom or molecule in a magnetic eld.
The positron follows a helical path along the magnetic eld with a small (5 m
radius) cyclotron orbit. Upon colliding with a test gas atom, the positron scatters at an angle , transferring some of its kinetic energy (which is initially in the
parallel component, Ek) into the perpendicular component, E?. The interaction
occurs on an atomic length scale (i.e., b 1 A). Since b is orders of magnitude
smaller than the radius of curvature of the positrons in the magnetic eld, the
positron scatters as if it were in a eld-free region. After the scattering event,
the positron continues to be guided by the magnetic eld, with some of its total
energy E transferred to the test atom and some into E?, depending on the angle
scattered and the elastic or inelastic nature of the scattering event.
36
Chapter 4
1.2
1.2
(b)
(a)
0.8
0.8
0.4
0.4
E⊥
E⊥ (eV)
elastic
0.0
0.0
0.4
0.8
0.0
1.2
0.0
1.2
0.4
0.8
1.2
1.2
(c)
(d)
0.8
0.4
0.0
0.0
0.8
inelastic
0.4
elastic
0.8
∆E
0.4
0.0
1.2
0.0
E||
E|| (eV)
inelastic
0.4
elastic
0.8
∆E
1.2
Figure 4.3: Simulated eects of elastic and inelastic scattering on the parallel and perpendicular energy components of an initially strongly magnetized, cold 1 eV charged
particle beam: (a) incident beam; (b) the eect of elastic scattering; (c) both elastic
and inelastic scattering; and (d) the scattered beam shown in (c), following an adiabatic
reduction of the magnetic eld by a factor of M = 10. The dashed lines show the
conservation of total energy as the positrons scatter through angle . The total energy
loss from a simulated inelastic collision is indicated by the shift E = 0:16 eV.
4.3.1 Elastic DCS Analysis
In cases where only elastic scattering is present (i.e., noble gases below the
threshold for electronic excitation and positronium formation), we can extract
Positron Scattering from Atoms and Molecules
37
the DCS from the parallel energy distribution of the scattered beam. Figure 4.3
shows a calculated energy distribution in (E?; Ek ) space for a cold 1 eV beam (a)
before, and (b) after an elastic scattering event. Because the collision is elastic,
the total positron energy E is conserved [Fig. 4.3(b) dashed line]; therefore
the scattering angle is determined solely by the amount of energy transferred
from Ek to E?. If we assume that the initial trajectory of the positron is in
the direction of the magnetic eld, then after an elastic scattering event, the
positron velocity, vk in the direction of the eld will be vk = v cos(), where v
is the total velocity of the positron, vk is the velocity along the magnetic eld,
and is the scattering angle. Thus, Ek = E cos2 (); which can be rewritten as:
q
= cos 1 Ek =E:
(4.1)
The assumption that the incoming positron trajectory is in the direction of the
magnetic eld is valid for Ek E?. Typically a positron will have Ek 1 eV
and E? 0:025 eV. This means that the positrons have an initial angular
spread [ = sin 1(v?=v)] of 9Æ , which provides an estimate of our angular
resolution.
In order to calculate the DCS, we need to know not only to which angle a
given Ek corresponds, but also how many positrons are scattered into that angle.
For a given applied voltage V0 , the RPA discriminates against all particles with
parallel energy components Ek less than eV0 , and so the RPA measures the
parallel energy distribution integrated over energies above Ek. Therefore the
integrated parallel energy distribution normalized to unity I (Ek ) measures the
probability for a positron to have a parallel energy greater than or equal to Ek .
Using this integrated energy distribution, the elastic DCS is obtained by
dE dI (E ) !
d
= C d
k dE k ;
(4.2)
d
k
where the constant of proportionality, C , relates the scattering cross-section to
the scattering probability, dEk =d
represents the relation between the eective
solid angle sampled and the energy increments used in the RPA measurement,
and dI (Ek )=dEk is the probability that a positron will be scattered into the
energy range dEk . The constant C is given by
1
C= = ;
(4.3)
Ps
nl
where is the cross section, Ps is the probability of a scattering event, n is
the number density of the test gas molecules, and l is the positron path length
through the scattering cell. For the experiment described here,
(4.4)
C (a20 ) = 0:029=Pav (torr);
38
Chapter 4
where Pav is the average pressure. We have assumed a scattering cell temperature of 20Æ C and a path length equal to the scattering cell length of 38 cm. The
quantity dEk =d
in Eq. (4.2) can be calculated using Eq. (4.1) to yield:
dEk
1 qEE :
=
(4.5)
k
d
Substituting Eq. (4.5) into Eq. (4.2), we obtain a nal relationship between the
RPA data I (Ek ) and the DCS for the elastic scattering event:
!
q
dI(E
)
d
k
0
= C EEk dE ;
(4.6)
d
k
where C 0 = C=. Using Eqs. (4.1) and (4.6) we can determine the DCS for a
positron-atomic or positron-molecular elastic scattering event given the parallel
energy distribution of the scattered beam.
4.3.2 Measurement of Total Inelastic Cross Sections
We are also able to measure total inelastic cross sections for positron-atom or
positron-molecule scattering. Figure 4.3(c) shows the simulated eects on E?
and Ek of both elastic and inelastic scattering from a positron beam. The
positrons that participate in an inelastic scattering event lose some energy E ,
transferring it to the atom or molecule, and are represented by the shifted beam
in Fig. 4.3(c). In order to measure the total inelastic cross section, we must
be able to distinguish between an inelastically scattered positron that has lost
E to the target atom and an elastically scattered positron whose scattering
angle corresponds to a loss of E in its parallel energy component. It is clear
from Fig. 4.3(c) that by simply measuring the parallel energy distribution of the
scattered beam we cannot distinguish between these two events. To circumvent
this problem, we take advantage of the adiabatic invariant E?=B , discussed in
Section 2.1, for a charged particle in a slowly varying magnetic eld of strength
B . If the positron scatters in a magnetic eld Bs and is guided adiabatically
into a lower eld Ba , where it is analyzed (see Fig. 4.1 magnets #2 and #3,
respectively), then E? is reduced by the ratio of the large eld to the small eld,
M = Bs =Ba , while the total energy of the positron is still conserved. For a large
reduction in the eld (M 1), the resulting parallel energy Ek is approximately
equal to the total energy E . Figure 4.3(d) shows the scattered beam after it has
undergone a reduction M = 10 in magnetic eld. It is clear from this gure that
the parallel energy distributions of the elastic and inelastic scattering events are
now well separated, and therefore they can be discriminated by the RPA.
4.3.3 Total Cross Sections
Although we have not yet published any data on positron total scattering cross
sections, such measurements are done routinely using magnetized positron beams
Positron Scattering from Atoms and Molecules
39
[43, 60, 100] and are, in principle, easy to do with the system described here.
The probability for a positron to undergo any scattering event (which is proportional to the total scattering probability) can be measured in two steps. First
the RPA is set to 0 V, and the total beam strength is measured. Then the
unscattered beam strength is measured by adjusting the RPA voltage a small
increment, V , below the beam energy. The analyzer rejects all positrons which
have either undergone an inelastic scatter with energy loss greater than eV or
have transferred parallel energy greater than eV into perpendicular energy by
elastic scattering, thus discriminating against all scattered positrons. The total scattering cross section is determined by comparing this signal to the total
beam strength and scaling it by the constant of proportionality C , derived in
Eq. (4.4), which relates the TCS to the scattering probability. By repeating this
at dierent values of beam energy, the total cross section as a function of beam
energy can be determined for a given atomic or molecular species.
4.4
Experimental Results
4.4.1 Dierential Cross Sections
Using the method described in Section 4.3.1, we have been able to make DCS
measurements for both Ar and Kr at energies ranging from 0.4 to 2.0 eV [31].
The data presented here is the rst experimental study of low-energy elastic scattering of positrons from noble gases. DCS measurements are important because
they oer a strict test of theoretical models, which in turn test our understanding of the processes governing positron scattering. These processes dier greatly
from their electron scattering counterparts. For example, in low-energy positronatomic elastic scattering, the two dominant long range interactions (static and
polarization), which are additive in the electron scattering case, tend to cancel
for positron scattering. Virtual positronium formation is also believed to play
an important role in positron-atom elastic scattering, and of course this process
is not available in the electron case [23].
The DCS data presented here were taken using our earlier accumulator,
which was not optimized for such measurements. Although the experimental
setup and operation are similar to that described in Section 4.2, the details of
the experiment are quite dierent and are described elsewhere [31]. Figure 4.4
shows the raw data, taken on the new scattering apparatus designed specically
for such experiments. The data are normalized to unity for a 1 eV positron beam
scattered from argon atoms. The open circles are the positron beam data with no
argon gas present. The closed circles are the scattered positron beam where the
pressure in the gas cell has been adjusted for approximately 10% total scattering.
Because the positron beam energy (1 eV) is below all inelastic processes, such as
positronium formation and electronic excitation, the scattering is purely elastic.
The bottom axis of Fig. 4.4 shows the applied voltage on the RPA. The upper
40
Chapter 4
angle (deg)
90
80 70
60
50
40
0.4
0.6
30
20 10
0
positrons (arb. units)
1
0
0.0
0.2
0.8
1.0
1.2
retarding potential (V)
Figure 4.4: RPA data for positron-argon elastic scattering: (Æ) 1 eV positron beam with
no argon gas present; and () positron beam after scattering from argon. The upper
horizontal axis indicates scattering angle corresponding to a given positron parallel
energy which is shown on the lower axis.
axis shows the corresponding scattering angle , as dened by Eq. (4.1), for a
given loss in the parallel energy of the positron. For example, a 1 eV positron
which scatters transferring 0.2 eV into E?, leaving 0.8 eV in Ek, corresponds to
a scattering angle of approximately 27Æ . Note that the upper axis shows only
scattering angles up to 90Æ . Positrons that are scattered greater than 90Æ , (i.e.
back-scattered) exit from the entrance of the gas cell. These back scattering
events are discussed in Section 4.5.
Figure 4.5 shows absolute DCS measurements in atomic units for positronkrypton scattering at energies of 1.0 and 2.0 eV. Our experiment simultaneously
collects both back-scattered and forward-scattered positrons [see Section 4.5],
Positron Scattering from Atoms and Molecules
30
30
(a)
25
dσ/dΩ (ao2)
41
20
20
15
15
10
10
5
5
0
0
0
20
40
60
(b)
25
80
0
angle (deg)
20
40
60
80
Figure 4.5: Absolute dierential elastic cross sections for positron-krypton scattering at
energies of (a) 1.0 and (b) 2.0 eV. Solid lines are theoretical predictions of McEachran,
[75], folded around = =2 [see Section 4.5]. There are no tted parameters.
et al.
and so the DCS data is plotted versus the scattering angle, folded around =
=2. We compare these data with the polarized orbital calculation by McEachran
et al. [75], where the polarized orbital is a perturbation of the bound-state wave
function due to the incident positron. There is good absolute agreement between
the DCS data and theory, which has also been folded around = =2, over the
entire range of energies and angles. For the energies of these measurements, the
theoretically predicted contribution due to back-scattering is negligible, so the
data represents mainly the forward-scattered positrons.
For the data taken on our earlier apparatus [i.e., Figs. 4.5, 4.6, and 4.8], the
dominant source of error is statistical uctuations due to low repetition rates.
This causes an uncertainty in the measurement of 20%. As described in
Section 4.2, our new scattering apparatus has a greatly improved repetition
rate (2 s vs. 20 s), which should improve this statistical error by 3 for a
data set taken in the same amount of time. Another source of error, which is
systematic in nature, is our ability to accurately measure the test-gas pressure.
This pressure is measured using a stable ion gauge located outside the scattering
cell. We extrapolate the pressure inside the cell using a particle code simulation
[5]. We believe that the combination of the external ion gauge and the particle
code provides an absolute pressure measurement better than 10%. In the new
scattering apparatus, this systematic error has been reduced to less than 1% by
42
Chapter 4
16
16
(b)
12
12
8
8
4
4
2
dσ/dΩ (ao )
(a)
0
0
0
20
40
60
80
16
0
20
40
60
80
16
(d)
(c)
12
12
8
8
4
4
0
0
0
20
40
60
80
0
20
40
60
80
angle (deg)
Figure 4.6: Dierential elastic cross sections for positron-argon scattering at 0.4, 0.7,
1.0 and 1.5 eV are shown in plots (a){(d) respectively. Solid and dotted lines are the
theoretical predictions of McEachran,
[74] and Dzuba,
[23] respectively. The
data and theory are folded around = =2 because the experiment does not distinguish
between forward-scattered and back-scattered positrons [see Section 4.5].
et al.
et al.
directly measuring the test gas pressure [see Section 4.2].
We also measured previously the DCS for positron scattering from argon,
which has a total scattering cross section roughly half that of krypton [31].
Figure 4.6 shows the absolute DCS for positron-argon scattering at energies
from 0.4 to 1.5 eV. The solid lines are the predictions of the polarized orbital
positrons (arb. units)
Positron Scattering from Atoms and Molecules
43
1
0.16 eV
0.16 eV
(a)
0
0.1
(b)
0.3
0.5
0.7 0.1
0.3
0.5
0.7
retarding potential (V)
Figure 4.7: RPA data for a positron-CF4 inelastic scattering event with a magnetic ratio
between the scattering cell and analyzer of M = 10 in (a) and M = 1 in (b). The open
circles correspond to a 0.55 eV positron beam with no CF4 present. The solid circles
with a spline t (solid line) correspond to the scattered beam following excitation of the
vibrational mode (3) in CF4 . The arrow indicates the 0.16 eV energy loss due to the
vibrational excitation.
calculations of McEachran et al. [74]. The dotted lines are the predictions of
a many-body theory by Dzuba et al. [23]. For the larger beam energies, 1.0
and 1.5 eV, there is good agreement between the experiment and theory at all
angles. However at lower beam energies, 0.4 and 0.7 eV, there is a systematic
disagreement between the data and predictions for large angles ( 60Æ ). We
believe that this is due to the eect of trapped positrons making multiple passes
through the scattering cell. A discussion of this eect and the method we have
developed to circumvent it is discussed in Section 4.5.
4.4.2 Total Inelastic Cross Sections
Using the cold beam, we measured the rst low-energy total vibrational cross section for positron-molecule scattering, studying the excitation of the vibrational
modes in CF4 [31]. The data in Fig. 4.7, which were taken using our new scattering apparatus, show the integrated parallel energy distribution of positron-CF4
scattering, with magnetic ratios (a) M = 10 and (b) M = 1 between the analyzer and scattering cell. The closed and open circles are measurements for a
0.55-eV positron beam with and without the CF4 test gas present. The arrows
correspond to an energy loss of 0:16 eV due to the excitation of a vibrational
vibrational cross section (ao2)
44
Chapter 4
10
x 1/5
8
6
4
2
0
0.0
0.2
0.4
0.6
0.8
1.0
energy (eV)
Figure 4.8: Inelastic cross section as a function of energy for the vibrational excitation
of CF4 () by positrons, and (Æ) from electron swarm data. The electron data is from
Ref. [85] and plotted at 1=5 actual value. The collisions with CF4 excite the asymmetric
stretch mode 3 at an energy of 0.157 eV.
mode in CF4. We have identied this energy loss to be due to the asymmetric
stretch mode 3 (0.157 eV). This is the dominant mode observed in both electron
scattering and infra-red absorption experiments [14,72], and it closely matches
the energy loss that we observe. It is clear from Fig. 4.7 that the application
of a magnetic ratio between the scattering cell and analyzer greatly reduces the
eect of elastic scattering on Ek , thus permitting accurate measurement of the
inelastic scattering cross section [see Section 4.3.2]. The probability of an inelastic scattering event is determined from Fig. 4.7(b) by measuring the magnitude
of the scattered component relative to that of the incident beam. The total
vibrational cross section is then calculated using the scale factor C in Eq. (4.4).
Figure 4.8 shows the inelastic cross-section as a function of beam energy for
positron and electron collisions with CF4. The data in Fig. 4.8 were taken using
our earlier scattering apparatus, which has a maximum magnetic ratio M = 3.
Although this ratio does increase the separation between elastic and inelastic
scattering events, it is not large enough to completely remove all elastic scattering
Positron Scattering from Atoms and Molecules
45
eects from the parallel energy distribution, and this represents a source of
systematic error in the measurements. We compare our data with the only
available electron vibrational cross-section measurements for CF4, which were
obtained using the swarm technique [85]. While the electron cross-section has a
distinct peak above the 3 threshold (0.157 eV), the positron data is qualitatively
dierent. This dierence raises potentially interesting theoretical questions [39].
Although there is theoretical work on the excitation of vibrational modes in
molecules by positrons [19, 28, 62], to our knowledge there are no theoretical
predictions for positron scattering from CF4 . The only other measurements
we are aware of for positron scattering from CF4 are the total cross-sections
measured above 1 eV by Mori et al. [82].
4.4.3 Recent Results
The work presented in Sections 4.4.1 and 4.4.2 represent some of the initial
experiments to study two-body positron-matter interactions in the energy region below 1 eV. From this work we learned a great deal about the strengths
and weaknesses of performing cold positron scattering experiments in a highly
magnetized system (see Section 4.5). Using our new scattering apparatus and accumulator we are beginning to exploit this knowledge to explore positron-atomic
and positron-molecular scattering processes in more detail. The following are
some recent results that we have obtained. They illustrate further the large
range of positron scattering processes that we can now investigate.
A number of recent experiments have focused on measuring positron-molecule
total vibrational cross sections. Figure 4.9 shows total inelastic cross sections
for positron excitation of the vibrational mode of CO at 0.266 eV. The data,
which was taken over the range of energies from 0.5 to 7 eV, are compared to the
results of (solid line) a body-xed vibrational close-coupling theory by Gianturco
et al. [29] and, (dotted line) earlier work by Jain [56] employing a model polarization potential. Note the excellent absolute agreement between the theoretical
predictions of Gianturco et al. and the data.
We have also measured total inelastic cross sections in cases where more
than one vibrational mode is observed. Figure 4.10 shows positron-CH4 total
vibrational cross sections at two excitation energies. The closed circles represent
the combined vibrational cross sections of the 2 and 4 vibrational modes which
occur at an energy of 0.190 and 0.162 eV, respectively (i.e., our present energy
resolution is not suÆcient to distinguish between the modes). Similarly, the
open circles represent the 1 and 3 vibrational modes at an energy of 0.362
and 0.374 eV. The separation between the 2 ,4 and 1,3 modes can be seen
clearly in the inset in Fig. 4.10 which shows the RPA data for 0.5 eV positronCH4 scattering events. It is important to note that the technique described in
Section 4.3.2 allows us to make these measurements, even though the vibrational
cross sections are 20 times smaller than the elastic scattering cross sections.
vibrational cross section (a02)
46
Chapter 4
2
1
0
0
2
4
6
energy (eV)
Figure 4.9: Inelastic cross section (in atomic units) as a function of energy for positron
excitation of the vibrational mode of CO at 0.266 eV. The solid and dotted lines are the
predictions of theories by Gianturco
[29] and Jain [56], respectively. There are no
tted parameters.
et al.
Figure 4.11 shows RPA data for a 0.5 eV positron beam scattering inelastically from CO2. The two steps in the data set are caused by the excitation of
the 2 bending mode and the 3 asymmetric stretch mode, which have energies
of 0.083 and 0.291 eV, respectively. We also have indications that the inelastic
cross section for the 1 symmetric stretch mode (which is much smaller than
the other two modes) can be measured for beam energies above 1 eV. Our
ability to measure the total vibration cross section for the 3 mode at 83 meV,
(i.e., 4 a20 at 0.5 eV), represents the lowest energy vibrational mode excited
by positron impact ever measured, and an example of what can be expected of
the cold positron beam technique.
Lastly, using the cold beam we have made total cross section measurements
at energies lower than any previous experiment. Figure 4.12 shows the total cross
sections for positron-molecule scattering in CF4, CH3 F, and CH4 at energies as
Positron Scattering from Atoms and Molecules
47
2
vibrational cross section (a0 )
4
3
1
signal
1.0
2
0.9
0.0
0.1
0.2
0.3
0.4
0.5
retarding potential (V)
0
0.5
0.6
0.7
0.8
energy (eV)
Figure 4.10: Inelastic cross section as a function of positron energy for the vibrational
excitation of the () 2 + 4, and (Æ) 1 + 3 modes of CH4. The inset shows an expanded
view of the RPA data for 0.5 eV positron-CH4 scattering events. The steps represent
the scattering cross section due to the excitation of the 2 + 4 modes at 0:18 eV and
the 1 + 3 modes at 0:37 eV.
low as 50 meV. By extending these measurements to more of the partially uorinated hydrocarbons, we hope to increase our understanding of the anomalously
large annihilation rates for these molecules [54]. The scattering experiments described above are a small sample of what we expect to accomplish in the future
using the cold positron beam.
48
Chapter 4
positrons (arb. units)
1.00
0.95
0.90
ν3 (0.291 eV), 10 a02
ν2 (0.083 eV), 4 a02
0.85
0.0
0.1
0.2
0.3
0.4
0.5
retarding potential (V)
Figure 4.11: RPA data for a 0.5 eV positron-CO2 inelastic scattering event. The two
inections represent the positron energy loss due to the excitation of the 2 and 3
modes. The solid line is a t to the data.
4.5
Measurements Using a Magnetic Beam { Further
Considerations
While scattering experiments in a highly magnetized system have some unique
advantages over experiments performed using an electrostatic beam, there are
also disadvantages to this approach. One diÆculty is the detection of backscattered positrons. In an electrostatic experiment this is relatively easy as long
as the experiment has the capability to move the detector beyond a 90Æ scattering
angle, which is usually possible except near 180Æ . In our system, the positrons
are forced to follow the magnetic eld after scattering. Figure 4.13(b) depicts
the path of a positron as it back-scatters at 150Æ from an atom or molecule. The
location of the positron accumulator, the scattering cell, and analyzer are shown
Positron Scattering from Atoms and Molecules
49
total cross section (a02)
300
200
100
0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
energy (eV)
Figure 4.12: Total cross sections for positron scattering with () CF4 , (Æ) CH3F, and
() CH4.
in Fig. 4.13(a). The horizontal arrows show the path of the positron through
the scattering apparatus, while the vertical arrows depict the energy transfer
from Ek into E? due to an elastic scattering event. When the positron backscatters, it transfers some of its parallel energy into perpendicular energy and
travels back out of the entrance of the scattering cell. Since it has lost some Ek ,
it is then reected by the potential barrier created by the positron accumulator
[see Fig. 4.13(b)] and travels back through the scattering cell, where it has a
90% probability of passing through the cell without scattering (since the single
pass scattering probability has been adjusted to be 10%). If the positron is not
scattered in the second pass through the cell, the RPA will detect the positron
as if it were scattered at 30Æ in the forward direction. Thus, we are unable to
distinguish between back-scattered and forward-scattered positrons, and so we
display the DCS results as a superposition of the two scattering components
folded around = =2. We are currently developing a technique to which will
50
Chapter 4
Figure 4.13: (a) Schematic diagram of the scattering experiment, showing the relative
positions of the positron accumulator, scattering cell and retarding
potential analyzer.
The lower gures show the path of a positron: (a) after a 150Æ scatter with the test
gas; and (b) after a 70Æ scatter. Vertical parts of the trajectories indicate a transfer of
energy from the parallel energy component Ek, into the perpendicular component E?,
due to elastic scattering events.
allow us to distinguish between the back-scattering and forward-scattering events
in order to obtain the full DCS (see Section 6.2.2).
Another eect of scattering in a magnetic eld was discovered through a
systematic discrepancy between our DCS data and theoretical predictions for
large angle scattering ( > 60Æ ) [see Section 4.4.1]. We noticed that when
the theory predicted appreciable scattering cross sections at large angles, our
experimental results were consistently low [see Fig. 4.6 (a) and (b)]. Figure
4.13(c) shows the motion of a positron as it scatters elastically at 70Æ . In this
scattering event, the positron transfers an appreciable amount of Ek into E?. If
Positron Scattering from Atoms and Molecules
51
the RPA is set up to discriminate against a 70Æ scattering event, the scattered
positron will be trapped in the potential well created by the analyzer and the
positron accumulator exit gate, as shown in Fig. 4.13(c). After each bounce,
the positron passes through the scattering cell potentially rescattering from the
test gas. If the positron rescatters, it can transfer some of its E? back into Ek ,
thereby allowing it to pass through the analyzer and be detected. This will have
the eect of making large angle scattering events look like small angle ones in
the DCS data. Since the positron must bounce back and forth a number of times
before it has a signicant chance of rescattering, these secondary scatters can be
eliminated by time resolving the measurement.
Such time-resolved measurements have been accomplished using a potential
barrier [not shown in Fig. 4.13(a)] located in front of the detector, to prevent
any trapped positrons which have rescattered from reaching the detector. The
potential barrier is raised after a few microseconds, which is enough time for
the initial scattered beam to pass the barrier, but is still short enough to block
any rescattered positrons. A typical bounce time for a trapped positron is a few
micro-seconds, and so in order to distinguish between the initial scattering event
and any rescattered positrons, the beam must have a pulse width less than this.
We have been able to create cold pulsed positron beams with a FWHM pulse
width less than 1 s, which are suitable for this purpose. Figure 4.14 shows
the eects of time resolution on a 0.7 eV positron-argon DCS. The open circles
and lled circles are data taken without and with time resolution, respectively.
Looking at the solid lines, which are tted to the data, one can see that time
resolution in the DCS measurement results in an increase in the measured large
angle scattering and a decrease in the small angle scattering, which is what one
would expect if the rescattered positrons were eliminated. Thus the data in Fig.
4.14 indicates that, by time resolving the data, we can eliminate the diÆculty
of secondary scattering at large angles.
Another complication in studying scattering using a highly magnetized system is the insensitive detection of scattering events near 90Æ . Equation (4.3)
shows that the probability of a scattering event is proportional to the positron's
path length. As we have discussed, a positron in a magnetic eld travels along
a helical path. Therefore, the path length of a positron traveling through the
scattering cell will be greater than the length of the cell. The ratio of these two
lengths depends only on the ratio of the energy components, E? and Ek,
lhelix =lk =
q
1 + E?=Ek ;
(4.7)
where lhelix is the path length of the positron and lk is the eective path length
for a positron moving in a straight line along the magnetic eld. For a cold
positron beam with Ek of 1 eV and E? of 0.025 eV, the ratio lhelix=lk is 1.01,
or a 1% correction to the straight path, which is negligible. Problems occur for
scattering events close to 90Æ . In this case, the positron transfers nearly all of its
52
Chapter 4
8
dσ/dΩ (ao2)
6
4
2
0
0
20
40
60
80
angle (deg)
Figure 4.14: Eects of time resolving the DCS measurement for 0.7 eV positron-argon
scattering: (Æ) data taken without time-resolution and () with time-resolution (t =
6 s). The solid lines are t to the data. Time resolution increases the detection
eÆciency for large angle scattering events.
parallel energy into the perpendicular component. It then moves slowly through
the cell, gyrating rapidly in the direction perpendicular to the magnetic eld
until it makes another scattering collision.
We can estimate the maximum angle at which single scattering can be assumed by requiring that the probability for a second scattering event be small.
On average, the positron will scatter in the center of the gas cell. Since we have
set the probability of scattering to 10% for a path length l equal to the scattering
cell length, we can increase the path length by a factor of 6 and still have an
acceptable probability for a second scatter of only 30%. Using Eqs. (4.1) and
(4.7), this implies that any initial scattering at less than 80Æ will have less than
a 30% chance of scattering a second time.
Positron Scattering from Atoms and Molecules
53
Some diÆculties are common to both the magnetized and electrostatic scattering experiments. An example is measurement of dierential scattering near
zero degrees. For the electrostatic case, the problem is caused by an inability to
physically locate the detector near the unscattered beam, which is necessary to
measure small-angle scattering. The diÆculty for the magnetized system is the
inability to separate the parallel energy distribution for small angle scattering
from that of the incident beam. This diÆculty is compounded by eects represented in Eq. (4.1), which relates the scattering angle to an energy transfer
out of Ek and into E?. One can see from this equation, which is plotted as the
upper horizontal axis in Fig. 4.4, that scattering angles below 10Æ are diÆcult
to study, even with our cold positron beam. This problem can, in principle, be
circumvented using a colder beam. Similar problems to those near 0Æ are encountered for scattering angles near 180Æ in both the magnetic and electrostatic
experiments
Conducting the experiments in a magnetic eld has some distinct advantages
and disadvantages compared to electrostatic scattering experiments. The distinct advantage is that we can conveniently use our state-of-the-art cold positron
beam, allowing us to study scattering in the range of energies E < 1 eV which
has previously been inaccessible to experiments. In addition, the advantages
include a large eective detector size (all scattered positrons are collected), no
moving parts, and a simple gas cell and analyzer design (with no need for a
complicated gas jet). Some of the disadvantages include a more complicated
analysis, dierential cross sections near 90Æ are diÆcult to detect, and measuring the DCS with both elastic and inelastic processes present is also diÆcult [see
Section 4.5].
4.6
Chapter Summary
We have begun to exploit the ability to produce a state-of-the-art cold positron
beam to study atomic and molecular physics in a new range of positron energies.
This eort has led us to investigate techniques to study scattering events in a
magnetic eld. We are continuing to improve our understanding of this type
of experiment. Even at this early stage of development, we have been able to
make new measurements in the energy range below 1 eV, including DCS measurements where even the equivalent electron experiments have proven diÆcult.
Examples of these include, positron-atom DCS for argon and krypton at energies
ranging from 0.4 to 2.0 eV. As discussed above, we have been able to make the
rst measurements of low-energy positron-molecule vibrational cross sections,
studying positron-CF4 collisions down to positron energies as low as 0.2 eV.
Currently, we are continuing our study of positron-molecular vibrations, by
measuring the total vibrational cross sections for a broad range of molecules
including CO, CO2, and CH4 , at energies from 0.5 up to 7 eV. We have also
54
Chapter 4
begun studying total cross sections for positron-molecule scattering and have
recently measured positron total scattering cross sections for CF4 , CH3, and
CH4 at energies from 50 meV to several electron volts. By continuing to push
the limits of low-energy positron scattering, we hope to study a broad range of
positron-matter interactions, not only in atomic and molecular systems but also
in materials and at material surfaces [37].
Chapter 5
The Electron-Positron
Beam-Plasma Instability
5.1
Introduction
Electron-positron plasmas are examples of a larger class of equal-mass plasmas
(or pair plasmas) that owe their unique properties to the symmetry between the
two oppositely charged species. Electron-positron plasmas in particular have
been extensively studied theoretically because of their relevance in astrophysical
contexts such as pulsar magnetospheres [76]. The linear properties of these
plasmas are well known [51,107,111]. Their nonlinear properties are currently
the focus of theoretical and numerical investigations [22,42,63,70,90,92,96{98,
112,113].
Studies of electron-positron plasmas in the laboratory present substantial
challenges to the experimentalist. Until recently, insuÆcient numbers of positrons
were available to create even single-component positron plasmas. With the introduction of a modied Penning-Malmberg trap for accumulating large numbers of
positrons [104], it became possible to conduct the rst electron-positron plasma
experiments in a beam-plasma system [34]. Unfortunately, the accumulation of
large numbers of positrons is possible only because of the outstanding connement properties of Penning traps [87], which can conne only one sign of charge.
The creation of an electron-positron plasma in the laboratory requires solving
the classic plasma physics problem of neutral plasma connement, and none of
the current congurations for conning neutral plasmas has suÆciently good
connement for electron-positron plasmas.
One possible approach to creating equal-mass plasmas is to use positive and
negative ions rather than electrons and positrons. Plasmas containing both positive and negative ions are relatively easy to create by the well-known method
of producing a hot-cathode discharge in a mixture of electronegative and electropositive gases such as sulfur hexauoride and argon. Positive ions are created
55
56
Chapter 5
by ionization of the argon and negative ions are created by electron attachment
to the sulfur hexauoride [50]. Unfortunately, for the experiments conducted
to date, there was a rather large mass ratio between the ions (140:40 for SF6
and Ar), and so these plasmas are not strictly equal-mass plasmas. Even if it
were possible to obtain a more nearly equal mass ratio by the judicious choice
of gases, a fundamental problem remains: it is currently impossible to entirely
eliminate the small residual electron component, which can completely alter the
properties of such plasmas, even in concentrations of less than 1%. Plasmas of
this type are therefore properly considered to be three-component plasmas.
Using another approach, Schermann and Major created an electron-free plasma consisting of positive and negative ions of (almost) equal mass in a Paul trap
by ionizing thallium iodide to create Tl+ and I ions [93]. The deconning eects
of RF heating were overcome by the cooling eect of a light buer gas, helium.
For electron-positron plasmas, this approach is much less attractive, because the
cooling eect of helium for light particles will be minimal. This approach may
work, however, by using a molecular species with a high inelastic cross section to
provide the required energy loss mechanism. On the basis of current knowledge
of positron-molecule collision cross sections, the vibrational excitation of carbon
tetrauoride is the most attractive candidate (see Section 4.4.2). In addition to
the Paul trap other approaches to conne equal-mass plasmas have also been
investigated, including the use of combined traps [108], magnetic mirrors [6],
and nested traps [36].
Nonetheless, all of these techniques are still relatively complicated, and were
not attempted for the experiments described in this chapter. Instead, we approached the experimental study of the electron-positron plasmas by investigating the electron-positron beam-plasma system. This approach involves transmitting the abundant species (the electrons) in a single pass through the scarce
species (the positrons) conned in a Penning trap. This permits us to take
advantage of the good connement properties of positron plasmas in Penning
traps, while still studying a two-component equal-mass system.
The beam-plasma system is interesting in its own right. The free energy in
the relative streaming of the two species can give rise to a variety of instabilities
and consequent plasma heating [99]. These eects can be important in a variety
of laboratory, magnetospheric and astrophysical plasmas. Beam-plasma eects
are currently being investigated theoretically for electron-positron plasmas in
the context of wave generation and particle acceleration [22, 42, 92, 112]. The
transit-time instability, which is the subject of this chapter, was rst studied
because of its potential as a source of microwave radiation [45,83].
The data described in this chapter represents a second generation beamplasma experiment. In our earlier beam-plasma experiments, an electron beam
formed from a hot cathode was transmitted through a positron plasma stored
in both a cylindrical and quadrupole Penning trap geometry. In the cylindrical
case, a two-stream instability, which caused strong heating of the plasma, was
The Electron-Positron Beam-Plasma Instability
57
studied. When the electron beam was passed through a positron plasma stored in
a quadrupole Penning trap geometry, the electron beam produced a transit-time
instability which excited the center of mass mode in the plasma. Because of the
large energy spread generated by the hot-cathode electron gun, both experiments
were restricted to studying the instabilities at energies above 1 eV. However,
theoretical predictions for both the cylindrical [26] and quadrupole [21] trap
geometries showed that the maximum growth rate and onset of the instability
should occur below this energy. It was clear, therefore, that the narrower energy
spread of the electron beam described in Section 5.2.2 would greatly improve
the ability to study the instabilities in the energy range near their onset.
This chapter discusses the rst application of this electron beam, using it
to further investigate the transit-time instability for a positron plasma stored
in a quadrupole well. New results are presented exploring the instability in
the previously unexplored energy range, including the low-energy onset of the
instability. These results are compared with the predictions of a new analytical
cold-uid theory that accurately models the system, and excellent agreement is
obtained. Future plans for beam-plasma experiments done both in the cylindrical
and quadrupole Penning traps are also discussed.
This chapter is structured as follows. In Sec. 5.2, the details of the experimental apparatus and techniques used to perform the experiments are presented.
In Sec. 5.3, the experimental results are discussed and compared with theory.
Finally, a summary is presented in Section 5.4.
5.2
Description of the Experiment
All of the data described in this chapter was taken on the earlier apparatus. Figure 5.1(a) shows a schematic drawing of the beam-plasma experiment. The apparatus consists of a cylindrical Penning-Malmberg trap coaxial with a quadrupole
Penning trap. Both traps are enclosed in the same vacuum vessel and use the
same conning magnetic eld. As shown in Fig. 5.1(a) and (b), the cylindrical
trap is the third stage of the three-stage accumulator.
The procedure for conducting the experiment can be split into three phases:
(1) Accumulation of a cold positron plasma, which is stored in the quadrupole
trap; (2) Accumulation of a reservoir of thermalized electrons stored in the
cylindrical trap, and the extraction of an electron beam from this reservoir; and
(3) measurement of the excitation of the transit-time instability in the positron
plasma by the electron beam. A more detailed explanation of these three phases
is presented below.
5.2.1 Positron Plasma Parameters
A plasma containing 1:6 107 positrons is accumulated in the cylindrical trap using the technique described in Section 2.4 and then shuttled into the quadrupole
58
Chapter 5
Figure 5.1: (a) Schematic diagram of the beam-plasma experiment showing the 22Na
positron source and moderator, movable hot-cathode electron source, magnetic eld coil,
imaging system, and trap electrode structures. (b) Expanded diagram of the cylindrical
and quadrupole Penning traps. Each trap consists of three electrodes from left to right,
referred to as the entrance-gate, dump, and exit-gate electrodes. The quadrupole trap
also has a small pickup electrode located near the positron plasma. (c) Schematic
diagram of the potential prole V(z) used to contain the electrons and positrons in
there respective potential wells.
59
density (arb. units)
The Electron-Positron Beam-Plasma Instability
0
1
2
3
radial position (cm)
Figure 5.2: Axially integrated radial density prole of a plasma containing 1:3 107
positrons (solid line). Also shown is the radial prole of 9the electron beam (dashed line),
which is formed from a cold plasma containing 1 10 electrons. As discussed in the
text these proles indicate a positron plasma diameter of Dp ' 2:4 cm and an electron
beam diameter of Db ' 0:4 cm.
trap at an eÆciency of 80%. The plasma cools to room temperature (0.025 eV)
in approximately 1 s by further collisions with the nitrogen buer gas [38]. The
buer gas is then pumped out to a base pressure of 5 10 9 torr in 10 s.
In order to conrm that the accumulated positrons are in the plasma state,
it is necessary to obtain plasma parameters such as the Debye length and the
dimensions of the charge cloud. These parameters were not measured directly,
but inferred from measurements of the axial-integrated plasma density prole.
This is accomplished by dumping the plasma onto a phosphor screen biased to
10 kV, which is then imaged using a CCD camera [see Fig. 5.1(a)]. The solid
line in Fig. 5.2 shows a typical axial-integrated radial density prole of a plasma
containing 1:3 107 positrons. The dashed line represents the narrower density
prole of the electron beam, which will be described in Sec. 5.2.2. The radial
60
Chapter 5
0.2
radial position (cm)
2
x 106 (cm-3)
0.3
0.4
0.5
1
0.6
(a)
0
number density (106 cm-3)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(b)
0.0
-2
-1
0
1
2
axial position (cm)
Figure 5.3: (a) Contour plot of the density of a 1:3 107 positron plasma stored in
the quadrupole well. It was calculated using a Poisson-Boltzmann equilibrium code,
using the radial density prole of the positrons shown in Fig. 5.2 as the initial positron
distribution. (b) Positron
number density along the trap axis, indicating a central
density of np ' 6 105 cm 3:
density prole of the plasma is used as input to a Poisson-Boltzmann equilibrium
code to calculate the spatial distribution of the positrons in the trap.
The results of this calculation are displayed in Fig. 5.3(a), which shows the
The Electron-Positron Beam-Plasma Instability
61
positron number density as a function of position in the plasma. From the gure
a plasma length of Lp ' 2:8 cm FWHM and a plasma diameter of Dp ' 3 cm
FWHM is determined. These dimensions corresponds to a plasma aspect ratio
of = Lp=Dp ' 0:9. We note that the plasma diameter calculated in this way
is slightly larger than the plasma diameter given in Fig. 5.2. This is due to the
line-integrated nature of the CCD camera image, versus the three dimensional
positron density prole in Fig. 5.3(a).
Figure 5.3(b) shows the positron number density as a function of position
along the axis of symmetry of the trap. The gure indicates a central plasma
density of np ' 6 105 cm 3, yielding a Debye length of D ' 1:5 mm. By
comparing the Debye length with the plasma dimensions, we see that D Lp,
D Dp , and ND 104 1, where ND is the number of positrons in a Debye
sphere. Thus, the charge cloud indeed satises all of the conditions to be a
single-component plasma.
The last parameter needed for the analysis of the experiment is the plasma
space charge potential. It is important to determine this parameter because it
eects the energy of the electron beam as it passes through the positron plasma
[see Section 5.2.3]. The space charge potential is determined by dumping the
plasma and measuring the potential dierence on the dump electrode at the
start and stop of the dump, as dened by the potential when the positrons start
exiting the trap and the potential when all the positrons have been expelled,
respectively. For the plasmas used in this experiment, a space charge of 1.6 V
is typically measured. In order to verify the direct measurement of the space
charge, the Poisson-Boltzmann code was used to calculate it. Figure 5.4 shows
the potential prole of the quadrupole trap along its axis of symmetry with
and without a positron plasma present. The dierence between the two curves
represents the positron plasma space charge of 1:3 V for a plasma containing
1:3 107 positrons. The discrepancy between the space charge measurement and
calculation is most likely caused by uncertainties in the measured total number
of positrons, which is needed in the Poisson-Boltzman code.
5.2.2 Cold Electron Beam Parameters
Following accumulation of a positron plasma in the quadrupole trap, a cold
electron beam is generated in the manner described in Section 3.4. To accurately
study the electron-beam positron-plasma transit-time instability, the rise time
of the electron beam current, tr must satisfy the condition tr 1= , where is
a typical growth rate of the instability ( 5 104 s 1), and the beam current
must remain constant over the duration of the interaction. We use the adaptive
dump technique (see Fig. 3.6) to produce a beam that satises these criteria.
It is also important that the beam diameter be small compared to the plasma
diameter. One simple way to accomplish this is by making the electron beam
source diameter small compared to the positron plasma, since the extracted
62
Chapter 5
potential (V)
0
-2
-4
-6
1.3 V
-8
-10
-10
-5
0
5
10
axial position (cm)
Figure 5.4: Potential prole along axis of symmetry of the trap without any charge
present (solid line), and with 1:3107 positrons (dashed line). The dierence in potential
between the two plots at the center is due to the plasma space charge of 1.3 V.
electron beam diameter is directly related to the size of the electron source.
To do this, a 2.9 mm aperture is located in front of the hot-cathode electron
gun. Figure 5.2 compares the radial proles of the electron beam and positron
plasma. The gure indicates an electron beam diameter Db ' 4 mm. Thus the
electron beam diameter is smaller than that of the plasma, and there is little
radial expansion during the beam formation process.
5.2.3 Beam-Plasma Experiment
The beam-plasma transit-time instability is studied in the following manner.
First, a positron plasma of 1:6 107 positrons is accumulated in the cylindrical
trap and transferred into the quadrupole trap. As the positron plasma cools
to room temperature, the hot-cathode electron gun is moved into position [see
Fig. 5.1(a)], and an electron plasma is accumulated in the cylindrical well. Figure 5.1(c) shows schematically the potential proles generated by the cylindrical
The Electron-Positron Beam-Plasma Instability
63
and quadrupole traps that are used to conne the electron and positron plasmas,
respectively, along with the space charge of both plasmas (indicated by shading). After the electron and positron plasmas have cooled to room temperature,
an electron beam is generated and magnetically guided through the positron
plasma. As the electron beam traverses the positron plasma, the transit-time
instability causes the center of mass oscillations to grow in amplitude. The
oscillations are detected using a pickup electrode, shown in Fig. 5.1(b), which
measures the image charge generated by the positron plasma.
After each cycle, the beam energy is adjusted by varying the potential on the
exit-gate electrode of the cylindrical trap. The energy of the electron beam relative to the positron plasma is denoted in Fig. 5.1(c) by E0 . The relative beam
energy is the sum of eVs and eVt , where Vs is the space charge of the positron
plasma, and Vt (which is negative in Fig. 5.1(c)) is the potential dierence between the dump electrode of the quadrupole trap and the exit-gate electrode of
the cylindrical trap. For example, if the dump and exit-gate electrodes of the
quadrupole and cylindrical traps were both at the same potential, the electron
beam energy through the positron plasma would be eVs.
As we discuss in the next section, in an earlier study of the transit-time
instability [34] the measured growth rates were an order of magnitude larger
than they are for the current experiment. Consequently, when the electron
beam passed through the plasma, any noise in the plasma could act as a \seed"
for the growth of the instability. In the current experiment, there are a few beam
energies where the growth rates are not large enough to insure that the center of
mass mode can grow above the detection amplier noise before the reservoir of
electrons is depleted. In these cases, we found it essential to actively \seed" the
center of mass mode to some small amplitude before the electron beam is turned
on. This is accomplished by applying a sinusoidal signal, at the same frequency
as the center of mass mode ( 4:2 MHz), to the entrance gate of the quadrupole
trap. Because of the high-Q properties of the quadrupole trap, the timing of
the seeding is not critical, and can be done as much as a few milliseconds before
the electron beam is turned on. To insure that the seeding processes does not
eect the measured growth rate a systematic check was performed by measuring
the instability growth rate with and without actively seeding the plasma in an
energy regime where the seeding was not necessary.
5.3
Results
Figure 5.5 shows the rms signal from the pickup electrode as the electron beam
passes through the positron plasma. The inset shows a 2:6 s long time record
of the pickup electrode signal, illustrating the individual oscillations at 4.2 MHz.
The plasma center of mass mode has an oscillation frequency corresponding to
that of a single positron oscillating in a quadrupole potential well. This frequency
64
Chapter 5
rms amplitude (arb. units)
1.0
0.8
0.6
0.4
4.2 MHZ
0.2
0.0
30
40
50
60
70
80
90
time (µs)
Figure 5.5: The rms amplitude from the pickup electrode signal, showing the growth
of the transit-time instability excited by an electron beam traversing a positron plasma
stored in a quadrupole Penning trap. The inset shows the pickup electrode signal on an
expanded time scale at 60 s, illustrating the individual center of mass oscillations at a
frequency of 4.2 MHz.
q
is given by fz = qV=42 mZ02, where q is the positron charge, V ' 15 V is the
potential dierence applied between the end and dump electrodes, m is the
positron mass, and Z0 ' 6:3 cm is a length parameter which is dened by the
quadrupole geometry [109]. Using this expression, a center of mass oscillation
frequency fz ' 4:1 MHz is predicted. The dierence between the calculated and
measured frequencies is likely due to small departures from the ideal quadrupole
geometry.
The electron beam is turned on at t = 0 exciting the instability in the plasma.
Some time later the amplitude of the center of mass oscillations begin to grow
out of the noise in the center of mass mode. After the initial linear growth, the
growth rate of the center of mass mode begin to decrease, causing the amplitude
The Electron-Positron Beam-Plasma Instability
65
of oscillation to overshoot its nal value and then eventually stabilize (not shown
in Fig. 5.5). In the earlier study of the transit-time instability [34], the initial
growth phase was also followed by a decay phase. Using a -ray detector, it was
determined that the decay in this case was due to the ejection of positrons from
the quadrupole trap when the plasma oscillations became so large that they
were no longer conned by the potential well. We note that the potential well
conning the positrons was 6 eV deep, and the electron beam energy used
was only 1 { 2 eV, so that some of the positrons must have been accelerated to
energies much larger than that of the relative beam energy. A particle-in-cell
simulation conrmed this, showing that the center of mass mode, could cause
the ejection of positrons from the trap [34].
In the present experiment, we decided to study the system in a less violent
regime. To accomplish this, beam currents were used that are an order of magnitude smaller than in the earlier experiment. This resulted in small amplitude
plasma oscillations. We veried using a -ray detector that these oscillations
do not eject positrons from the potential well, and therefore the overshoot and
saturation of the center of mass mode must be caused by an eect other than
positron ejection. The most likely possibility is that non-linear plasma eects are
responsible for the observed saturation. We have not yet carried out a detailed
study of the saturation, but believe that it is likely to be an interesting area of
further research.
Figure 5.6 shows the measured instability growth rate as a function of beam
energy for two beam currents. The predictions of an analytic cold-uid theory,
developed by Dubin [21], are also shown. This theory, which has no tted
parameters, is in excellent agreement with the data over the entire range of
energies studied. The agreement with the cold-uid theory implies that the
system acts like a transit time oscillator, exciting a high-Q oscillation of the
center of mass motion of the plasma. The theory also predicts that the instability
growth rates scale linearly with beam currents, which is conrmed by the data
sets taken with beam currents of 0.03 and 0.08 A. Our previous experiments,
which also agreed with the cold-uid theory, were limited by large beam energy
spreads to studying the instability at beam energies above the maximum of the
growth rate. The new cold electron beam allows us to study the instability over
the entire range of interest; from onset through the maximum growth rate and
beyond. We note that the data indicates the onset of the instability occurring
at an energy 0.05 eV below the theoretically predicted onset. This discrepancy
is most likely due to the nite electron beam energy spread of 0.1 eV, which is
not included in the theoretical calculations.
66
Chapter 5
10
growth rate (104 s-1)
8
6
0.03 µA
4
2
0
0.0
0.4
0.8
1.2
1.6
2.0
energy (eV)
Figure 5.6: Growth rates for the beam-plasma interaction in the quadrupole trap. The
solid and open circles are the data using a 0.03 and 0.08 A electron beam, respectively.
The solid lines are the results of a cold-uid theory by Dubin [21]. There are no tted
parameters used between the theory and experiment.
5.4
Chapter Summary
We have investigated a beam-plasma system in a new and interesting range of
beam energies, studying a transit-time instability in a positron plasma stored
in a quadrupole Penning trap. Using a new technique that we developed to
generate cold electron beams, we have been able to measure the growth rate of
the center of mass mode from the onset of the instability through the maximum
in the growth rate. Comparison of the data with the results of a cold-uid theory
by Dubin [21] shows excellent agreement over the entire range of energies and
beam currents studied.
The experiments presented here represent further studies of the electronpositron beam-plasma system. These experiments are the only ones currently
The Electron-Positron Beam-Plasma Instability
67
being conducted on this type of plasma, or on any type of equal-mass plasma.
Although the techniques described here are not suitable for studying the properties of equal-mass plasmas where the species have no net currents relative to
each other, it is likely to be the only experimentally accessible system available
for studies of this type of plasma in the near term. The excellent agreement
between the experimental results and an analytical cold-uid theory is therefore
gratifying. Experiments of this type are likely to be of value in extending our
understanding of equal-mass plasmas. The combination of the ability to eÆciently accumulate and store large numbers of positrons with the new technique
to produce cold electron beams should provide opportunities to investigate other
interesting phenomena in electron-positron plasmas.
68
Chapter 5
Chapter 6
Conclusion
6.1
Summary
The ability to generate a versatile beam of intense, cold positrons has opened up a
new range of energies for the experimental study of positron-matter interactions.
The cold beam, which is discussed in Chapter 3, is formed from a thermalized
reservoir of cold positrons stored in a Penning trap. The beam has an energy
spread of 18 meV and can be generated with a beam energy ranging from <
50 meV upward, making it the only tool available for the study of positronmatter interactions signicantly below 1 eV. We have also been able to generate
cold electron beams using this technique. Although not as exotic an application
as for positrons, the generation of a cold electron beam in a strong magnetic eld
is a non-trivial task, and allows us to peruse experimental studies of electronpositron plasmas where a strong eld is necessary for positron connement.
Using the cold positron beam we have begun to study low-energy positronatomic and positron-molecular interactions (see Chapter 4). The rst positronmolecule vibrational excitation cross section measurements were performed using
the original positron accumulator by studying positron excitation of the 3 vibrational mode of CF4 at energies down to the onset of the vibrational mode.
Positron-atom DCS at energies below that of any previous measurement have
also been studied. These measurements, which included positron excitation of
argon and krypton at energies from 0.4 to 2.0 eV were compared with theoretical
predictions [23,74,75] and agree well over a large range of energies and angles.
Using information gained from these experiments and an improved understanding of positron scattering in a strong magnetic eld, a new scattering apparatus was designed and constructed. This apparatus, which is now attached to
the new positron accumulator, has a greatly improved signal-to-noise ratio, data
collection rate, and magnetic ratio, allowing the measurement of positron scattering cross sections in a fraction of the time required in the earlier apparatus.
By performing scattering experiments on the new apparatus, we have expanded
69
70
Chapter 6
our understanding of the strengths and limitations of positron scattering in a
strong magnetic eld, and expect that future design enhancements will expand
further our ability to study low-energy positron-matter interactions.
Currently a broad study of positron-molecule vibrational excitation cross
sections is being performed. Studies include the vibrational excitation of CO,
CO2, and CH4 with energies ranging from 0.5 up to 7 eV. Multiple vibrational
modes have been measured in a number of these molecules. For example, in
CO2, we have measured the vibrational cross sections for the 3 and 2 modes
which occur at 291 and 83 meV, respectively. The ability to perform these
state-of-the-art vibrational cross section measurements is, in turn, a motivation
for future theoretical calculations. Initial comparison of the measured positronCO2 inelastic cross sections with theoretical predictions by Gianturco [29] are
in excellent agreement over almost the entire energy range calculated, and plans
to calculate vibrational cross sections for the other molecules being studied are
in progress [27].
The rst application of the method developed to produce a cold magnetized electron beam was used to extend our understanding of electron-beam
positron-plasma instabilities. Chapter 5 discusses growth rate measurements of
the instability caused by transmitting the cold electron beam through a positron
plasma stored in a quadrupole well. Using the cold beam we have been able to
extend the range of energies in which the transit-time instability can be studied
to include the onset through the maximum in growth rate. Comparison of the
measured instability growth rates with predictions from a cold uid theory [21]
are in excellent absolute agreement over the entire range of energies and beam
currents studied.
6.2
Future Work
6.2.1 Cold Beams
Because the techniques used to form the cold positron and electron beams are
not fully matured, it is reasonable to expect that modications of the beam
formation method will yield improvements in the energy spread, intensity, and
versatility of the beam. We are currently constructing a high-eld (5 tesla) ultrahigh-vacuum positron storage stage [103]. In this device, the positron plasmas
will be surrounded by electrodes cooled to 4 kelvin, and so the plasma will
equilibrate to this temperature by cyclotron radiation. Using this accumulator
as a reservoir of cold positrons, in principle, it should be possible to produce
extremely bright milli-electron-volt positron beams for use in a broad range of
experiments.
Another experiment under investigation is the ability to compress the positron
plasma using a rotating electric eld. It has been shown for electrons that by
applying a rotating electric asymmetry onto the walls of the conning electrodes,
Conclusion
71
a torque can be applied to the plasma, thereby compressing it [2]. A decrease
in the plasma radius by a factor 4:5, accompanied by an increase in the
plasma density of 20, was observed in these experiments. Use of this technique
in the high-eld stage should increase the plasma density and the brightness of
extracted positron beams.
Lastly, there are two approaches being considered to improve the overall
eÆciency of the positron accumulator, and therefore the signal-to-noise ratio of
our positron-matter experiments. The rst is to increase the trapping eÆciency
of the new positron accumulator. The current eÆciency of the accumulator is
still 50% less than that of the earlier accumulator. Indications point to a reduced
positron life time through the second stage, caused by positron transport to the
walls. By enlarging the diameter of the second stage electrode, the trapping
eÆciency is hoped to improve to at least 30% from its current value of 20%. The
second approach is to increase the eÆciency of our neon moderator by changing
the cone geometry. The cone geometry used in the new accumulator was designed
to maximize the number of the high energy positrons from the source that can
hit it. Unfortunately, the eÆciency of the new cone geometry is 5 times worse
than the cone geometry used on the earlier accumulator. We expect that by
changing the cone geometry back to one that more closely resembles the original
design, the moderator eÆciency should improve by a factor of 5.
6.2.2 Positron Atomic and Molecular Scattering
The low-energy atomic and molecular scattering experiments described in this
thesis are examples of scattering processes that can be studied using the new
apparatus. Plans to systematically measure total inelastic cross sections for
positron-molecule interactions are already underway. We have also shown that
these techniques can be used to measure total cross sections at energies as low
as 50 meV. By measuring the total cross section for partially uorinated hydrocarbons (i.e., CH4 through CF4), we hope to improve our understanding of
the anomalous large annihilation rates observed in positron-hydrocarbon interactions [54].
Another interesting related topic to study is the possible existence and nature of low-lying positron-molecule resonances and weakly bound states. For
example, by measuring the elastic scattering cross section in the regime where
ka < 1, where k is the momentum of the positron and a is the s-wave scattering
length, it should be possible to use asymptotic formulae to measure the sign and
magnitude of the scattering length, and thereby determine the expected energies
of bound states or resonances [39].
Possible techniques to measure the scattering cross section, (E0 ; E; ) when
both elastic and inelastic processes are present are also being investigated. Here,
E0 and E are the incident and scattered positron energies and is the scattering
angle. This can, in principal, be accomplished by taking retarding-potential
72
Chapter 6
data at dierent magnetic ratios M , and using analysis techniques similar to
tomographic reconstruction to unfold the scattering cross section [7].
One of the current deciencies of the scattering experiments described here is
the inability to distinguish back-scattered particles from forward-scattered particles (see Section 4.5). One possible solution to this problem is to incorporate
a set of E B drift plates in between the scattering cell and accumulator. By
placing appropriate potentials on the E B drift plates, the positron path can
be altered depending on the direction of travel along the magnetic eld. Therefore, by placing E B plates and an energy analyzer between the scattering cell
and the accumulator, the energy distribution for both the back-scattered and
forward-scattered positrons could be measured. This would then allow measurement of the complete DCS from nearly 0 to 180 degrees. For total inelastic
cross sections, this technique would allow separate measurements of the contribution due to the forward and backward inelastic scattering cross sections,
providing information about the angular dependence of the inelastic scattering
cross section.
The study of surfaces is also an interesting area of future research. There
are a number of novel surface science techniques, such as Positron Annihilation
Lifetime Spectroscopy (PALS), which can be enhanced using the cold positron
beam [37]. In PALS positrons injected into the surface are trapped and subsequently annihilate in vacancy-type defects. By measuring the positron lifetime,
information about the defects can be obtained. Using the cold beam can enhance
this technique, allowing a depth prole of the void size and concentration. This
is accomplished by varying the beam energy, and therefore the depth at which
the positron is implanted into the solid.
Lastly, although a broad survey of annihilation rates using room-temperature
positrons has been completed [53{55], there is only a small body of work on
positron annihilation as a function of positron energy below the threshold for
positron formation [66]. The cold beam could be used to greatly expand these
studies. For example, the parameter Ee+ Ei + EPs is the energy dierence
between the positron energy, Ee+ , and the positronium formation threshold
energy, Ei EPs , and it is thought to play a signicant role in annihilation
rates. Recent theories predict resonance behavior in the annihilation rates at
energies Ee+ Ei + EPs 0, and this eect could be studied experimentally
using the cold positron beam.
6.2.3 Electron-Positron Plasmas
There are a number of experiments that could now be performed to increase
our understanding of the electron-beam positron-plasma system, beyond those
described in Chapter 5. For example, by stabilizing the center of mass mode
with an active feedback system [106], it should be possible to measure the growth
rate of higher order modes. There may also be eects worth studying at energies
Conclusion
73
below the instability onset. In this energy range, cold-uid theory predicts that
there is a band of energies at which the growth rate is negative, and this should
therefore have a damping eect on the center of mass oscillations. By exciting
the center of mass mode to large amplitudes before the electron beam traverses
the plasma, we can, in principal, study these negative growth rates.
Also re-examining the two-stream instability generated in the cylindrical
trap [34] using the cold electron beam will likely yield interesting new insights.
It is clear from the earlier experiments that the onset of the instability occurs at
an energy below that studied. It should be possible to use the new cold beam to
measure the heating rate for a range of positron energies from the onset of the
instability through the maximum heating rate. Another interesting aspect of the
two-stream instability is the method in which the plasma is heated. The heating
is assumed to arise from the growth of unstable plasma modes which then transfer energy to the plasma particles. Direct measurements of these unstable modes
should prove insightful. To measure them, a pickup electrode must be located
near the plasma in order to detect the image charge generated by the (presumably short wavelength) plasma oscillations. We believe that these experiments
can oer further insight into the underlying physics of the instability.
Prospects for studying relativistic electron-positron plasmas, which is of keen
interest to the astrophysics community, continue to be poor, at least in the intermediate term. Obtaining Debye lengths that are smaller than the plasma size,
which is essential for the charge cloud to behave as a plasma, would require about
ve orders of magnitude more positrons than can now be accumulated. However, progress in the accumulation of positron plasmas continues. For example,
the number of positrons now available is three orders of magnitude more than
the rst positron plasma created in 1989, and a further two orders of magnitude
increase can be expected within the next few years [102].
6.3
Concluding Remarks
We have developed a new technique to produce intense, cold, magnetized positron
and electron beams. Using these beams we have studied both electron-beam
positron-plasma instabilities and two-body interactions of low-energy positrons
with atoms and molecules in an energy regime previously inaccessible to experiment. This research has expanding our knowledge of positron-matter interactions at low-energies and has helped to improve our ability to perform such
experiments in a magnetic eld. The work presented here demonstrates the eectiveness of this approach; we expect that the cold beam technique will continue
to make advances in both positron-atomic and beam-plasma physics.
74
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